diff --git a/src/analysis.lagda.md b/src/analysis.lagda.md
index 8790cf7fd6..a608189df4 100644
--- a/src/analysis.lagda.md
+++ b/src/analysis.lagda.md
@@ -3,6 +3,8 @@
 ```agda
 module analysis where
 
+open import analysis.absolute-convergence-series-real-banach-spaces public
+open import analysis.absolute-convergence-series-real-numbers public
 open import analysis.additive-complete-metric-abelian-groups-real-banach-spaces public
 open import analysis.complete-metric-abelian-groups public
 open import analysis.convergent-series-complete-metric-abelian-groups public
diff --git a/src/analysis/absolute-convergence-series-real-banach-spaces.lagda.md b/src/analysis/absolute-convergence-series-real-banach-spaces.lagda.md
new file mode 100644
index 0000000000..b85da50b37
--- /dev/null
+++ b/src/analysis/absolute-convergence-series-real-banach-spaces.lagda.md
@@ -0,0 +1,203 @@
+# Absolute convergence of series in real Banach spaces
+
+```agda
+module analysis.absolute-convergence-series-real-banach-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.convergent-series-real-banach-spaces
+open import analysis.convergent-series-real-numbers
+open import analysis.series-real-banach-spaces
+open import analysis.series-real-numbers
+
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequality-natural-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.coproduct-types
+open import foundation.dependent-pair-types
+open import foundation.function-types
+open import foundation.identity-types
+open import foundation.propositional-truncations
+open import foundation.propositions
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import linear-algebra.real-banach-spaces
+open import linear-algebra.sums-of-finite-sequences-of-elements-real-banach-spaces
+
+open import metric-spaces.cauchy-sequences-metric-spaces
+
+open import order-theory.large-posets
+
+open import real-numbers.cauchy-sequences-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.distance-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.rational-real-numbers
+```
+
+</details>
+
+## Idea
+
+A [series](analysis.series-real-banach-spaces.md) `Σ aₙ` in a
+[real Banach space](linear-algebra.real-banach-spaces.md) is said to
+{{#concept "absolutely converge" WDID=Q332465 WD="absolute convergence" Agda=is-absolutely-convergent-prop-series-ℝ-Banach-Space Disambiguation="series in a real Banach space"}}
+if the series of norms `Σ ∥aₙ∥` is a
+[convergent series](analysis.convergent-series-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  (σ : series-ℝ-Banach-Space V)
+  where
+
+  is-absolutely-convergent-prop-series-ℝ-Banach-Space : Prop (lsuc l1)
+  is-absolutely-convergent-prop-series-ℝ-Banach-Space =
+    is-convergent-prop-series-ℝ (map-norm-series-ℝ-Banach-Space V σ)
+
+  is-absolutely-convergent-series-ℝ-Banach-Space : UU (lsuc l1)
+  is-absolutely-convergent-series-ℝ-Banach-Space =
+    type-Prop is-absolutely-convergent-prop-series-ℝ-Banach-Space
+```
+
+## Properties
+
+### A Cauchy modulus for the sequence of partial sums of norms of terms in a series is a Cauchy modulus for the series
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  (σ : series-ℝ-Banach-Space V)
+  (M :
+    is-cauchy-sequence-ℝ
+      ( partial-sum-series-ℝ (map-norm-series-ℝ-Banach-Space V σ)))
+  where
+
+  abstract
+    neighborhood-add-cauchy-modulus-partial-sum-norm-series-ℝ-Banach-Space :
+      (ε : ℚ⁺) (n k : ℕ) → leq-ℕ (pr1 (M ε)) n →
+      neighborhood-ℝ-Banach-Space
+        ( V)
+        ( ε)
+        ( partial-sum-series-ℝ-Banach-Space V σ (n +ℕ k))
+        ( partial-sum-series-ℝ-Banach-Space V σ n)
+    neighborhood-add-cauchy-modulus-partial-sum-norm-series-ℝ-Banach-Space
+      ε n k με≤n =
+      let
+        open inequality-reasoning-Large-Poset ℝ-Large-Poset
+      in
+        chain-of-inequalities
+        dist-ℝ-Banach-Space V
+          ( partial-sum-series-ℝ-Banach-Space V σ (n +ℕ k))
+          ( partial-sum-series-ℝ-Banach-Space V σ n)
+        ≤ map-norm-ℝ-Banach-Space V
+            ( partial-sum-series-ℝ-Banach-Space V
+              ( drop-series-ℝ-Banach-Space V n σ)
+              ( k))
+          by
+            leq-eq-ℝ
+              ( ap
+                ( map-norm-ℝ-Banach-Space V)
+                ( inv (partial-sum-drop-series-ℝ-Banach-Space V n σ k)))
+        ≤ partial-sum-series-ℝ
+            ( map-norm-series-ℝ-Banach-Space V
+              ( drop-series-ℝ-Banach-Space V n σ))
+            ( k)
+          by
+            triangle-inequality-norm-sum-fin-sequence-type-ℝ-Banach-Space
+              ( V)
+              ( k)
+              ( _)
+        ≤ ( partial-sum-series-ℝ
+            ( map-norm-series-ℝ-Banach-Space V σ)
+            ( n +ℕ k)) -ℝ
+          ( partial-sum-series-ℝ (map-norm-series-ℝ-Banach-Space V σ) n)
+          by
+            leq-eq-ℝ
+              ( partial-sum-drop-series-ℝ
+                ( n)
+                ( map-norm-series-ℝ-Banach-Space V σ)
+                ( k))
+        ≤ dist-ℝ
+            ( partial-sum-series-ℝ
+              ( map-norm-series-ℝ-Banach-Space V σ)
+              ( n +ℕ k))
+            ( partial-sum-series-ℝ
+              ( map-norm-series-ℝ-Banach-Space V σ)
+              ( n))
+          by leq-diff-dist-ℝ _ _
+        ≤ real-ℚ⁺ ε
+          by
+            leq-dist-neighborhood-ℝ
+              ( ε)
+              ( _)
+              ( _)
+              ( pr2
+                ( M ε)
+                ( n +ℕ k)
+                ( n)
+                ( transitive-leq-ℕ (pr1 (M ε)) n (n +ℕ k) (leq-add-ℕ n k) με≤n)
+                ( με≤n))
+
+    is-cauchy-partial-sum-is-cauchy-partial-sum-norm-series-ℝ-Banach-Space :
+      is-cauchy-sequence-Metric-Space
+        ( metric-space-ℝ-Banach-Space V)
+        ( partial-sum-series-ℝ-Banach-Space V σ)
+    is-cauchy-partial-sum-is-cauchy-partial-sum-norm-series-ℝ-Banach-Space =
+      is-cauchy-sequence-modulus-neighborhood-add-sequence-Metric-Space
+        ( metric-space-ℝ-Banach-Space V)
+        ( partial-sum-series-ℝ-Banach-Space V σ)
+        ( pr1 ∘ M)
+        ( neighborhood-add-cauchy-modulus-partial-sum-norm-series-ℝ-Banach-Space)
+```
+
+### If a series is absolutely convergent, it is convergent
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  (σ : series-ℝ-Banach-Space V)
+  where
+
+  is-convergent-is-absolutely-convergent-series-ℝ-Banach-Space :
+    is-absolutely-convergent-series-ℝ-Banach-Space V σ →
+    is-convergent-series-ℝ-Banach-Space V σ
+  is-convergent-is-absolutely-convergent-series-ℝ-Banach-Space (lim-Σnorm , H) =
+    let
+      open
+        do-syntax-trunc-Prop
+          ( is-convergent-prop-series-ℝ-Banach-Space V σ)
+    in do
+      cauchy-mod ←
+        exists-cauchy-modulus-has-limit-sequence-ℝ
+          ( partial-sum-series-ℝ (map-norm-series-ℝ-Banach-Space V σ))
+          ( lim-Σnorm , H)
+      is-convergent-is-cauchy-sequence-partial-sum-series-ℝ-Banach-Space
+        ( V)
+        ( σ)
+        ( is-cauchy-partial-sum-is-cauchy-partial-sum-norm-series-ℝ-Banach-Space
+          ( V)
+          ( σ)
+          ( cauchy-mod))
+```
+
+## See also
+
+- [Absolute convergence of series of real numbers](analysis.absolute-convergence-series-real-numbers.md)
+
+## External links
+
+- [Absolute convergence of series in Banach spaces](https://en.wikipedia.org/wiki/Absolute_convergence#Proof_that_any_absolutely_convergent_series_in_a_Banach_space_is_convergent)
+  on Wikipedia
diff --git a/src/analysis/absolute-convergence-series-real-numbers.lagda.md b/src/analysis/absolute-convergence-series-real-numbers.lagda.md
new file mode 100644
index 0000000000..309cbb81ea
--- /dev/null
+++ b/src/analysis/absolute-convergence-series-real-numbers.lagda.md
@@ -0,0 +1,80 @@
+# Absolute convergence of series in the real numbers
+
+```agda
+module analysis.absolute-convergence-series-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import analysis.absolute-convergence-series-real-banach-spaces
+open import analysis.convergent-series-real-banach-spaces
+open import analysis.convergent-series-real-numbers
+open import analysis.series-real-banach-spaces
+open import analysis.series-real-numbers
+
+open import foundation.propositions
+open import foundation.universe-levels
+
+open import linear-algebra.real-banach-spaces
+```
+
+</details>
+
+## Idea
+
+A [series](analysis.series-real-numbers.md) `Σ aₙ` of
+[real numbers](real-numbers.dedekind-real-numbers.md) is said to
+{{#concept "absolutely converge" WDID=Q332465 WD="absolute convergence" Agda=is-absolutely-convergent-prop-series-ℝ Disambiguation="series of real numbers"}}
+if the series of absolute values `Σ |aₙ|`
+[converges](analysis.convergent-series-real-numbers.md).
+
+## Definition
+
+```agda
+module _
+  {l : Level}
+  (σ : series-ℝ l)
+  where
+
+  is-absolutely-convergent-prop-series-ℝ : Prop (lsuc l)
+  is-absolutely-convergent-prop-series-ℝ =
+    is-convergent-prop-series-ℝ (map-abs-series-ℝ σ)
+
+  is-absolutely-convergent-series-ℝ : UU (lsuc l)
+  is-absolutely-convergent-series-ℝ =
+    type-Prop is-absolutely-convergent-prop-series-ℝ
+```
+
+## Properties
+
+### If a series of real numbers is absolutely convergent, it is convergent
+
+```agda
+module _
+  {l : Level}
+  (σ : series-ℝ l)
+  where
+
+  is-convergent-is-absolutely-convergent-series-ℝ :
+    is-absolutely-convergent-series-ℝ σ →
+    is-convergent-series-ℝ σ
+  is-convergent-is-absolutely-convergent-series-ℝ H =
+    is-convergent-real-is-convergent-real-banach-space-ℝ
+      ( term-series-ℝ σ)
+      ( is-convergent-is-absolutely-convergent-series-ℝ-Banach-Space
+        ( real-banach-space-ℝ l)
+        ( series-terms-ℝ-Banach-Space
+          ( real-banach-space-ℝ l)
+          ( term-series-ℝ σ))
+        ( H))
+```
+
+## See also
+
+- [Absolute convergence of series in real Banach spaces](analysis.absolute-convergence-series-real-banach-spaces.md)
+
+## External links
+
+- [Absolute convergence](https://en.wikipedia.org/wiki/Absolute_convergence) on
+  Wikipedia
diff --git a/src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md b/src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md
index a2f53f1cf5..756e56a460 100644
--- a/src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md
+++ b/src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md
@@ -14,6 +14,7 @@ open import analysis.series-complete-metric-abelian-groups
 open import foundation.dependent-pair-types
 open import foundation.inhabited-types
 open import foundation.propositions
+open import foundation.subtypes
 open import foundation.universe-levels
 
 open import metric-spaces.cauchy-sequences-complete-metric-spaces
@@ -55,6 +56,11 @@ module _
 
   is-convergent-series-Complete-Metric-Ab : UU (l1 ⊔ l2)
   is-convergent-series-Complete-Metric-Ab = is-convergent-series-Metric-Ab σ
+
+convergent-series-Complete-Metric-Ab :
+  {l1 l2 : Level} (G : Complete-Metric-Ab l1 l2) → UU (l1 ⊔ l2)
+convergent-series-Complete-Metric-Ab G =
+  type-subtype (is-convergent-prop-series-Complete-Metric-Ab G)
 ```
 
 ## Properties
diff --git a/src/analysis/convergent-series-real-banach-spaces.lagda.md b/src/analysis/convergent-series-real-banach-spaces.lagda.md
index b553a4c38d..f790c49f26 100644
--- a/src/analysis/convergent-series-real-banach-spaces.lagda.md
+++ b/src/analysis/convergent-series-real-banach-spaces.lagda.md
@@ -10,16 +10,32 @@ module analysis.convergent-series-real-banach-spaces where
 open import analysis.additive-complete-metric-abelian-groups-real-banach-spaces
 open import analysis.convergent-series-complete-metric-abelian-groups
 open import analysis.convergent-series-metric-abelian-groups
+open import analysis.convergent-series-real-numbers
 open import analysis.series-real-banach-spaces
+open import analysis.series-real-numbers
 
 open import foundation.dependent-pair-types
+open import foundation.function-types
 open import foundation.inhabited-types
+open import foundation.logical-equivalences
 open import foundation.propositions
+open import foundation.subtypes
 open import foundation.universe-levels
 
+open import linear-algebra.normed-real-vector-spaces
 open import linear-algebra.real-banach-spaces
 
+open import lists.sequences
+
 open import metric-spaces.cauchy-sequences-metric-spaces
+open import metric-spaces.limits-of-sequences-metric-spaces
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.distance-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.sums-of-finite-sequences-of-real-numbers
+
+open import univalent-combinatorics.standard-finite-types
 ```
 
 </details>
@@ -56,6 +72,11 @@ module _
 
   is-convergent-series-ℝ-Banach-Space : UU (l1 ⊔ l2)
   is-convergent-series-ℝ-Banach-Space = is-convergent-series-Metric-Ab σ
+
+convergent-series-ℝ-Banach-Space :
+  {l1 l2 : Level} (V : ℝ-Banach-Space l1 l2) → UU (l1 ⊔ l2)
+convergent-series-ℝ-Banach-Space V =
+  type-subtype (is-convergent-prop-series-ℝ-Banach-Space V)
 ```
 
 ## Properties
@@ -80,6 +101,45 @@ module _
       ( σ)
 ```
 
+### A series in the real Banach space of real numbers converges if and only if its corresponding series of real numbers converges
+
+```agda
+module _
+  {l : Level}
+  (σ : sequence (ℝ l))
+  where
+
+  is-convergent-real-is-convergent-real-banach-space-ℝ :
+    is-convergent-series-ℝ-Banach-Space
+      ( real-banach-space-ℝ l)
+      ( series-terms-ℝ-Banach-Space (real-banach-space-ℝ l) σ) →
+    is-convergent-series-ℝ (series-terms-ℝ σ)
+  is-convergent-real-is-convergent-real-banach-space-ℝ (lim , is-lim) =
+    ( lim ,
+      preserves-limits-sequence-isometry-Metric-Space
+        ( metric-space-Normed-ℝ-Vector-Space (normed-real-vector-space-ℝ l))
+        ( metric-space-ℝ l)
+        ( id , λ d x y → inv-iff (neighborhood-iff-leq-dist-ℝ d x y))
+        ( λ n → sum-fin-sequence-ℝ n (σ ∘ nat-Fin n))
+        ( lim)
+        ( is-lim))
+
+  is-convergent-real-banach-space-is-convergent-ℝ :
+    is-convergent-series-ℝ (series-terms-ℝ σ) →
+    is-convergent-series-ℝ-Banach-Space
+      ( real-banach-space-ℝ l)
+      ( series-terms-ℝ-Banach-Space (real-banach-space-ℝ l) σ)
+  is-convergent-real-banach-space-is-convergent-ℝ (lim , is-lim) =
+    ( lim ,
+      preserves-limits-sequence-isometry-Metric-Space
+        ( metric-space-ℝ l)
+        ( metric-space-Normed-ℝ-Vector-Space (normed-real-vector-space-ℝ l))
+        ( id , λ d x y → neighborhood-iff-leq-dist-ℝ d x y)
+        ( λ n → sum-fin-sequence-ℝ n (σ ∘ nat-Fin n))
+        ( lim)
+        ( is-lim))
+```
+
 ### If a series converges, there exists a modulus making its partial sums a Cauchy sequence
 
 ```agda
diff --git a/src/analysis/convergent-series-real-numbers.lagda.md b/src/analysis/convergent-series-real-numbers.lagda.md
index 16804c6e49..1226c36539 100644
--- a/src/analysis/convergent-series-real-numbers.lagda.md
+++ b/src/analysis/convergent-series-real-numbers.lagda.md
@@ -12,6 +12,7 @@ open import analysis.convergent-series-metric-abelian-groups
 open import analysis.series-real-numbers
 
 open import foundation.propositions
+open import foundation.subtypes
 open import foundation.universe-levels
 
 open import real-numbers.cauchy-sequences-real-numbers
@@ -48,6 +49,9 @@ module _
 
   is-convergent-series-ℝ : UU (lsuc l)
   is-convergent-series-ℝ = is-convergent-series-Metric-Ab σ
+
+convergent-series-ℝ : (l : Level) → UU (lsuc l)
+convergent-series-ℝ l = type-subtype (is-convergent-prop-series-ℝ {l})
 ```
 
 ## Properties
diff --git a/src/analysis/series-real-banach-spaces.lagda.md b/src/analysis/series-real-banach-spaces.lagda.md
index dc7d5b3651..12f7c30508 100644
--- a/src/analysis/series-real-banach-spaces.lagda.md
+++ b/src/analysis/series-real-banach-spaces.lagda.md
@@ -9,7 +9,13 @@ module analysis.series-real-banach-spaces where
 ```agda
 open import analysis.metric-abelian-groups-normed-real-vector-spaces
 open import analysis.series-metric-abelian-groups
+open import analysis.series-real-numbers
 
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.natural-numbers
+
+open import foundation.function-types
+open import foundation.identity-types
 open import foundation.universe-levels
 
 open import linear-algebra.real-banach-spaces
@@ -54,3 +60,52 @@ module _
     series-ℝ-Banach-Space → sequence (type-ℝ-Banach-Space V)
   partial-sum-series-ℝ-Banach-Space = partial-sum-series-Metric-Ab
 ```
+
+## Properties
+
+### The series of norms
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  (σ : series-ℝ-Banach-Space V)
+  where
+
+  map-norm-series-ℝ-Banach-Space : series-ℝ l1
+  map-norm-series-ℝ-Banach-Space =
+    series-terms-ℝ (map-norm-ℝ-Banach-Space V ∘ term-series-ℝ-Banach-Space V σ)
+```
+
+### Dropping terms from a series
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  where
+
+  drop-series-ℝ-Banach-Space :
+    ℕ → series-ℝ-Banach-Space V → series-ℝ-Banach-Space V
+  drop-series-ℝ-Banach-Space = drop-series-Metric-Ab
+```
+
+### The partial sums of a series after dropping terms
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  where
+
+  abstract
+    partial-sum-drop-series-ℝ-Banach-Space :
+      (n : ℕ) (σ : series-ℝ-Banach-Space V) (i : ℕ) →
+      partial-sum-series-ℝ-Banach-Space V
+        ( drop-series-ℝ-Banach-Space V n σ)
+        ( i) ＝
+      diff-ℝ-Banach-Space V
+        ( partial-sum-series-ℝ-Banach-Space V σ (n +ℕ i))
+        ( partial-sum-series-ℝ-Banach-Space V σ n)
+    partial-sum-drop-series-ℝ-Banach-Space = partial-sum-drop-series-Metric-Ab
+```
diff --git a/src/analysis/series-real-numbers.lagda.md b/src/analysis/series-real-numbers.lagda.md
index 6b7455caa6..74b2105aae 100644
--- a/src/analysis/series-real-numbers.lagda.md
+++ b/src/analysis/series-real-numbers.lagda.md
@@ -10,12 +10,26 @@ module analysis.series-real-numbers where
 open import analysis.series-complete-metric-abelian-groups
 open import analysis.series-metric-abelian-groups
 
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.natural-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.function-types
+open import foundation.identity-types
+open import foundation.propositions
 open import foundation.universe-levels
 
 open import lists.sequences
 
+open import order-theory.increasing-sequences-posets
+
+open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-additive-group-of-real-numbers
+open import real-numbers.nonnegative-real-numbers
 ```
 
 </details>
@@ -42,3 +56,69 @@ term-series-ℝ = term-series-Metric-Ab
 partial-sum-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)
 partial-sum-series-ℝ {l} = partial-sum-series-Metric-Ab
 ```
+
+## Properties
+
+### The proposition that the terms of a series are nonnegative
+
+```agda
+is-nonnegative-prop-series-ℝ : {l : Level} → series-ℝ l → Prop l
+is-nonnegative-prop-series-ℝ σ =
+  Π-Prop ℕ (λ n → is-nonnegative-prop-ℝ (term-series-ℝ σ n))
+
+is-nonnegative-series-ℝ : {l : Level} → series-ℝ l → UU l
+is-nonnegative-series-ℝ σ = type-Prop (is-nonnegative-prop-series-ℝ σ)
+```
+
+### If the terms of a series of real numbers are nonnegative, the partial sums are monotonic
+
+```agda
+abstract
+  is-increasing-partial-sum-is-nonnegative-term-series-ℝ :
+    {l : Level} (σ : series-ℝ l) →
+    is-nonnegative-series-ℝ σ →
+    is-increasing-sequence-Poset (ℝ-Poset l) (partial-sum-series-ℝ σ)
+  is-increasing-partial-sum-is-nonnegative-term-series-ℝ {l} σ H =
+    is-increasing-leq-succ-sequence-Poset
+      ( ℝ-Poset l)
+      ( partial-sum-series-ℝ σ)
+      ( λ n → leq-left-add-real-ℝ⁰⁺ _ (term-series-ℝ σ n , H n))
+```
+
+### The series of absolute values
+
+```agda
+module _
+  {l : Level}
+  (σ : series-ℝ l)
+  where
+
+  map-abs-series-ℝ : series-ℝ l
+  map-abs-series-ℝ = series-terms-ℝ (abs-ℝ ∘ term-series-ℝ σ)
+```
+
+### Dropping terms from a series
+
+```agda
+module _
+  {l : Level}
+  where
+
+  drop-series-ℝ : ℕ → series-ℝ l → series-ℝ l
+  drop-series-ℝ = drop-series-Metric-Ab
+```
+
+### The partial sums of a series after dropping terms
+
+```agda
+module _
+  {l : Level}
+  where
+
+  abstract
+    partial-sum-drop-series-ℝ :
+      (n : ℕ) (σ : series-ℝ l) (i : ℕ) →
+      partial-sum-series-ℝ (drop-series-ℝ n σ) i ＝
+      partial-sum-series-ℝ σ (n +ℕ i) -ℝ partial-sum-series-ℝ σ n
+    partial-sum-drop-series-ℝ = partial-sum-drop-series-Metric-Ab
+```
diff --git a/src/linear-algebra.lagda.md b/src/linear-algebra.lagda.md
index 895b8bacdc..1e582c5ef5 100644
--- a/src/linear-algebra.lagda.md
+++ b/src/linear-algebra.lagda.md
@@ -51,6 +51,8 @@ open import linear-algebra.scalar-multiplication-tuples public
 open import linear-algebra.scalar-multiplication-tuples-on-rings public
 open import linear-algebra.seminormed-real-vector-spaces public
 open import linear-algebra.subsets-left-modules-rings public
+open import linear-algebra.sums-of-finite-sequences-of-elements-normed-real-vector-spaces public
+open import linear-algebra.sums-of-finite-sequences-of-elements-real-banach-spaces public
 open import linear-algebra.symmetric-bilinear-forms-real-vector-spaces public
 open import linear-algebra.transposition-matrices public
 open import linear-algebra.tuples-on-commutative-monoids public
diff --git a/src/linear-algebra/normed-real-vector-spaces.lagda.md b/src/linear-algebra/normed-real-vector-spaces.lagda.md
index 42c3f8902a..9f7b406225 100644
--- a/src/linear-algebra/normed-real-vector-spaces.lagda.md
+++ b/src/linear-algebra/normed-real-vector-spaces.lagda.md
@@ -32,8 +32,10 @@ open import metric-spaces.metrics
 open import metric-spaces.metrics-of-metric-spaces
 
 open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
@@ -207,6 +209,15 @@ module _
     nonnegative-dist-Seminormed-ℝ-Vector-Space
       ( seminormed-vector-space-Normed-ℝ-Vector-Space)
 
+  triangular-norm-Normed-ℝ-Vector-Space :
+    (v w : type-Normed-ℝ-Vector-Space) →
+    leq-ℝ
+      ( map-norm-Normed-ℝ-Vector-Space (add-Normed-ℝ-Vector-Space v w))
+      ( map-norm-Normed-ℝ-Vector-Space v +ℝ map-norm-Normed-ℝ-Vector-Space w)
+  triangular-norm-Normed-ℝ-Vector-Space =
+    triangular-seminorm-Seminormed-ℝ-Vector-Space
+      ( seminormed-vector-space-Normed-ℝ-Vector-Space)
+
   is-extensional-norm-Normed-ℝ-Vector-Space :
     (v : type-Normed-ℝ-Vector-Space) →
     map-norm-Normed-ℝ-Vector-Space v ＝ raise-ℝ l1 zero-ℝ →
@@ -384,6 +395,23 @@ module _
                   ( w)))))
 ```
 
+### The norm of the zero vector is zero
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : Normed-ℝ-Vector-Space l1 l2)
+  where
+
+  abstract
+    is-zero-norm-zero-Normed-ℝ-Vector-Space :
+      map-norm-Normed-ℝ-Vector-Space V (zero-Normed-ℝ-Vector-Space V) ＝
+      raise-ℝ l1 zero-ℝ
+    is-zero-norm-zero-Normed-ℝ-Vector-Space =
+      is-zero-seminorm-zero-Seminormed-ℝ-Vector-Space
+        ( seminormed-vector-space-Normed-ℝ-Vector-Space V)
+```
+
 ## See also
 
 - [Real Banach spaces](linear-algebra.real-banach-spaces.md), normed real vector
diff --git a/src/linear-algebra/real-banach-spaces.lagda.md b/src/linear-algebra/real-banach-spaces.lagda.md
index bdcc30ac25..60e48f9003 100644
--- a/src/linear-algebra/real-banach-spaces.lagda.md
+++ b/src/linear-algebra/real-banach-spaces.lagda.md
@@ -9,7 +9,10 @@ module linear-algebra.real-banach-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.positive-rational-numbers
+
 open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
@@ -69,6 +72,15 @@ module _
   metric-space-ℝ-Banach-Space =
     metric-space-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
 
+  type-ℝ-Banach-Space : UU l2
+  type-ℝ-Banach-Space =
+    type-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
+
+  neighborhood-ℝ-Banach-Space :
+    ℚ⁺ → type-ℝ-Banach-Space → type-ℝ-Banach-Space → UU l1
+  neighborhood-ℝ-Banach-Space =
+    neighborhood-Metric-Space metric-space-ℝ-Banach-Space
+
   is-complete-metric-space-ℝ-Banach-Space :
     is-complete-Metric-Space metric-space-ℝ-Banach-Space
   is-complete-metric-space-ℝ-Banach-Space = pr2 V
@@ -77,10 +89,6 @@ module _
   complete-metric-space-ℝ-Banach-Space =
     ( metric-space-ℝ-Banach-Space , is-complete-metric-space-ℝ-Banach-Space)
 
-  type-ℝ-Banach-Space : UU l2
-  type-ℝ-Banach-Space =
-    type-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
-
   map-norm-ℝ-Banach-Space : type-ℝ-Banach-Space → ℝ l1
   map-norm-ℝ-Banach-Space =
     map-norm-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
@@ -109,6 +117,21 @@ module _
   has-limit-cauchy-sequence-ℝ-Banach-Space =
     has-limit-cauchy-sequence-Complete-Metric-Space
       ( complete-metric-space-ℝ-Banach-Space)
+
+  dist-ℝ-Banach-Space : (u v : type-ℝ-Banach-Space) → ℝ l1
+  dist-ℝ-Banach-Space =
+    dist-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
+
+  commutative-dist-ℝ-Banach-Space :
+    (u v : type-ℝ-Banach-Space) →
+    dist-ℝ-Banach-Space u v ＝ dist-ℝ-Banach-Space v u
+  commutative-dist-ℝ-Banach-Space =
+    commutative-dist-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
+
+  diff-ℝ-Banach-Space :
+    type-ℝ-Banach-Space → type-ℝ-Banach-Space → type-ℝ-Banach-Space
+  diff-ℝ-Banach-Space =
+    diff-Normed-ℝ-Vector-Space normed-vector-space-ℝ-Banach-Space
 ```
 
 ## Properties
diff --git a/src/linear-algebra/sums-of-finite-sequences-of-elements-normed-real-vector-spaces.lagda.md b/src/linear-algebra/sums-of-finite-sequences-of-elements-normed-real-vector-spaces.lagda.md
new file mode 100644
index 0000000000..e5f32535a6
--- /dev/null
+++ b/src/linear-algebra/sums-of-finite-sequences-of-elements-normed-real-vector-spaces.lagda.md
@@ -0,0 +1,94 @@
+# Sums of finite sequences of elements in normed real vector spaces
+
+```agda
+module linear-algebra.sums-of-finite-sequences-of-elements-normed-real-vector-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.natural-numbers
+
+open import foundation.function-types
+open import foundation.universe-levels
+
+open import group-theory.sums-of-finite-sequences-of-elements-abelian-groups
+
+open import linear-algebra.normed-real-vector-spaces
+
+open import lists.finite-sequences
+
+open import order-theory.large-posets
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.sums-of-finite-sequences-of-real-numbers
+
+open import univalent-combinatorics.standard-finite-types
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "sum" Disambiguation="of a finite sequence of elements of a normed vector space over ℝ" Agda=sum-fin-sequence-type-Normed-ℝ-Vector-Space}}
+of a [finite sequence](lists.finite-sequences.md) of elements of a
+[normed vector space](linear-algebra.normed-real-vector-spaces.md) over the
+[real numbers](real-numbers.dedekind-real-numbers.md) is the
+[sum of the sequence](group-theory.sums-of-finite-sequences-of-elements-abelian-groups.md)
+in the [abelian group](group-theory.abelian-groups.md) of the vector space under
+addition.
+
+## Definition
+
+```agda
+sum-fin-sequence-type-Normed-ℝ-Vector-Space :
+  {l1 l2 : Level} (V : Normed-ℝ-Vector-Space l1 l2) (n : ℕ) →
+  fin-sequence (type-Normed-ℝ-Vector-Space V) n → type-Normed-ℝ-Vector-Space V
+sum-fin-sequence-type-Normed-ℝ-Vector-Space V =
+  sum-fin-sequence-type-Ab (ab-Normed-ℝ-Vector-Space V)
+```
+
+## Properties
+
+### The norm of the sum of a finite sequence is at most the sum of the norms of the terms of the sequence
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : Normed-ℝ-Vector-Space l1 l2)
+  where
+
+  abstract
+    triangle-inequality-norm-sum-fin-sequence-type-Normed-ℝ-Vector-Space :
+      (n : ℕ) (σ : fin-sequence (type-Normed-ℝ-Vector-Space V) n) →
+      leq-ℝ
+        ( map-norm-Normed-ℝ-Vector-Space V
+          ( sum-fin-sequence-type-Normed-ℝ-Vector-Space V n σ))
+        ( sum-fin-sequence-ℝ n (map-norm-Normed-ℝ-Vector-Space V ∘ σ))
+    triangle-inequality-norm-sum-fin-sequence-type-Normed-ℝ-Vector-Space 0 σ =
+      leq-eq-ℝ (is-zero-norm-zero-Normed-ℝ-Vector-Space V)
+    triangle-inequality-norm-sum-fin-sequence-type-Normed-ℝ-Vector-Space
+      (succ-ℕ n) σ =
+      let
+        open inequality-reasoning-Large-Poset ℝ-Large-Poset
+      in
+        chain-of-inequalities
+          map-norm-Normed-ℝ-Vector-Space V
+            ( sum-fin-sequence-type-Normed-ℝ-Vector-Space V (succ-ℕ n) σ)
+          ≤ ( map-norm-Normed-ℝ-Vector-Space V
+              ( sum-fin-sequence-type-Normed-ℝ-Vector-Space V
+                ( n)
+                ( σ ∘ inl-Fin n))) +ℝ
+            ( map-norm-Normed-ℝ-Vector-Space V (σ (neg-one-Fin n)))
+            by triangular-norm-Normed-ℝ-Vector-Space V _ _
+          ≤ sum-fin-sequence-ℝ (succ-ℕ n) (map-norm-Normed-ℝ-Vector-Space V ∘ σ)
+            by
+              preserves-leq-right-add-ℝ _ _ _
+                ( triangle-inequality-norm-sum-fin-sequence-type-Normed-ℝ-Vector-Space
+                  ( n)
+                  ( σ ∘ inl-Fin n))
+```
diff --git a/src/linear-algebra/sums-of-finite-sequences-of-elements-real-banach-spaces.lagda.md b/src/linear-algebra/sums-of-finite-sequences-of-elements-real-banach-spaces.lagda.md
new file mode 100644
index 0000000000..ba3aa4cac9
--- /dev/null
+++ b/src/linear-algebra/sums-of-finite-sequences-of-elements-real-banach-spaces.lagda.md
@@ -0,0 +1,67 @@
+# Sums of finite sequences of elements in real Banach spaces
+
+```agda
+module linear-algebra.sums-of-finite-sequences-of-elements-real-banach-spaces where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.natural-numbers
+
+open import foundation.function-types
+open import foundation.universe-levels
+
+open import linear-algebra.real-banach-spaces
+open import linear-algebra.sums-of-finite-sequences-of-elements-normed-real-vector-spaces
+
+open import lists.finite-sequences
+
+open import real-numbers.inequality-real-numbers
+open import real-numbers.sums-of-finite-sequences-of-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "sum" Disambiguation="of a finite sequence of elements of a real Banach space" Agda=sum-fin-sequence-type-ℝ-Banach-Space}}
+of a [finite sequence](lists.finite-sequences.md) of elements of a
+[real Banach space](linear-algebra.real-banach-spaces.md) is the
+[sum of the sequence](group-theory.sums-of-finite-sequences-of-elements-abelian-groups.md)
+in the [abelian group](group-theory.abelian-groups.md) of the space under
+addition.
+
+## Definition
+
+```agda
+sum-fin-sequence-type-ℝ-Banach-Space :
+  {l1 l2 : Level} (V : ℝ-Banach-Space l1 l2) →
+  (n : ℕ) → fin-sequence (type-ℝ-Banach-Space V) n → type-ℝ-Banach-Space V
+sum-fin-sequence-type-ℝ-Banach-Space V =
+  sum-fin-sequence-type-Normed-ℝ-Vector-Space
+    ( normed-vector-space-ℝ-Banach-Space V)
+```
+
+## Properties
+
+### The norm of the sum of a finite sequence is at most the sum of the norms of the terms of the sequence
+
+```agda
+module _
+  {l1 l2 : Level}
+  (V : ℝ-Banach-Space l1 l2)
+  where
+
+  abstract
+    triangle-inequality-norm-sum-fin-sequence-type-ℝ-Banach-Space :
+      (n : ℕ) (σ : fin-sequence (type-ℝ-Banach-Space V) n) →
+      leq-ℝ
+        ( map-norm-ℝ-Banach-Space V
+          ( sum-fin-sequence-type-ℝ-Banach-Space V n σ))
+        ( sum-fin-sequence-ℝ n (map-norm-ℝ-Banach-Space V ∘ σ))
+    triangle-inequality-norm-sum-fin-sequence-type-ℝ-Banach-Space =
+      triangle-inequality-norm-sum-fin-sequence-type-Normed-ℝ-Vector-Space
+        ( normed-vector-space-ℝ-Banach-Space V)
+```
diff --git a/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md b/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
index 71c916ccf9..c1219e7c5d 100644
--- a/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/cauchy-sequences-metric-spaces.lagda.md
@@ -9,6 +9,7 @@ module metric-spaces.cauchy-sequences-metric-spaces where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.archimedean-property-positive-rational-numbers
@@ -772,6 +773,56 @@ module _
       is-cauchy-sequence-pair-cauchy-sequence-Metric-Space)
 ```
 
+### To show a sequence `aₙ` is Cauchy, it suffices to have a modulus function such that for any `ε`, `aₙ` and `aₙ₊ₖ` are in an `ε`-neighborhood for sufficiently large `n`
+
+```agda
+module
+  _
+  {l1 l2 : Level}
+  (X : Metric-Space l1 l2)
+  (a : sequence-type-Metric-Space X)
+  (μ : ℚ⁺ → ℕ)
+  where
+
+  abstract
+    is-cauchy-modulus-neighborhood-add-sequence-Metric-Space :
+      ( (ε : ℚ⁺) (n k : ℕ) → leq-ℕ (μ ε) n →
+        neighborhood-Metric-Space X ε (a (n +ℕ k)) (a n)) →
+      (ε : ℚ⁺) →
+      is-cauchy-modulus-sequence-Metric-Space X a ε (μ ε)
+    is-cauchy-modulus-neighborhood-add-sequence-Metric-Space H ε p q με≤p με≤q =
+      let
+        lemma :
+          (m n : ℕ) → leq-ℕ (μ ε) m → leq-ℕ m n →
+          neighborhood-Metric-Space X ε (a n) (a m)
+        lemma m n με≤m m≤n =
+          let
+            ( k , k+m=n) = subtraction-leq-ℕ m n m≤n
+          in
+            tr
+              ( λ p → neighborhood-Metric-Space X ε (a p) (a m))
+              ( commutative-add-ℕ m k ∙ k+m=n)
+              ( H ε m k με≤m)
+      in
+        rec-coproduct
+          ( λ p≤q →
+            symmetric-neighborhood-Metric-Space
+              ( X)
+              ( ε)
+              ( a q)
+              ( a p)
+              ( lemma p q με≤p p≤q))
+          ( lemma q p με≤q)
+          ( linear-leq-ℕ p q)
+
+  is-cauchy-sequence-modulus-neighborhood-add-sequence-Metric-Space :
+    ( (ε : ℚ⁺) (n k : ℕ) → leq-ℕ (μ ε) n →
+      neighborhood-Metric-Space X ε (a (n +ℕ k)) (a n)) →
+    is-cauchy-sequence-Metric-Space X a
+  is-cauchy-sequence-modulus-neighborhood-add-sequence-Metric-Space H ε =
+    ( μ ε , is-cauchy-modulus-neighborhood-add-sequence-Metric-Space H ε)
+```
+
 ## See also
 
 - [Cauchy sequences in complete metric spaces](metric-spaces.cauchy-sequences-complete-metric-spaces.md)
diff --git a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
index 2cfda0f25e..984330b2f5 100644
--- a/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
+++ b/src/metric-spaces/limits-of-sequences-metric-spaces.lagda.md
@@ -31,6 +31,7 @@ open import foundation.universe-levels
 open import lists.sequences
 
 open import metric-spaces.cartesian-products-metric-spaces
+open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-functions-metric-spaces
 open import metric-spaces.sequences-metric-spaces
@@ -383,6 +384,46 @@ module _
     map-is-inhabited short-map-limit-modulus-sequence-Metric-Space
 ```
 
+### Isometries between metric spaces preserve limits
+
+```agda
+module _
+  {la la' lb lb' : Level}
+  (A : Metric-Space la la')
+  (B : Metric-Space lb lb')
+  (f : isometry-Metric-Space A B)
+  (u : sequence-type-Metric-Space A)
+  (lim : type-Metric-Space A)
+  where
+
+  isometry-map-limit-modulus-sequence-Metric-Space :
+    limit-modulus-sequence-Metric-Space A u lim →
+    limit-modulus-sequence-Metric-Space
+      ( B)
+      ( map-sequence
+        ( map-isometry-Metric-Space A B f)
+        ( u))
+      ( map-isometry-Metric-Space A B f lim)
+  isometry-map-limit-modulus-sequence-Metric-Space =
+    short-map-limit-modulus-sequence-Metric-Space
+      ( A)
+      ( B)
+      ( short-isometry-Metric-Space A B f)
+      ( u)
+      ( lim)
+
+  preserves-limits-sequence-isometry-Metric-Space :
+    is-limit-sequence-Metric-Space A u lim →
+    is-limit-sequence-Metric-Space
+      ( B)
+      ( map-sequence
+        ( map-isometry-Metric-Space A B f)
+        ( u))
+      ( map-isometry-Metric-Space A B f lim)
+  preserves-limits-sequence-isometry-Metric-Space =
+    map-is-inhabited isometry-map-limit-modulus-sequence-Metric-Space
+```
+
 ### If two sequences have limits in metric spaces, their pairing has a limit in the product space
 
 The converse has yet to be proved.
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 8210a60b2e..06ea8bb3f3 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -107,6 +107,7 @@ open import real-numbers.strict-inequality-nonnegative-real-numbers public
 open import real-numbers.strict-inequality-positive-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
 open import real-numbers.subsets-real-numbers public
+open import real-numbers.sums-of-finite-sequences-of-real-numbers public
 open import real-numbers.suprema-families-real-numbers public
 open import real-numbers.totally-bounded-subsets-real-numbers public
 open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers public
diff --git a/src/real-numbers/cauchy-sequences-real-numbers.lagda.md b/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
index a7d81021a9..8a716b033f 100644
--- a/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
@@ -9,6 +9,8 @@ module real-numbers.cauchy-sequences-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import foundation.dependent-pair-types
+open import foundation.inhabited-types
 open import foundation.universe-levels
 
 open import lists.sequences
@@ -20,6 +22,7 @@ open import metric-spaces.cauchy-sequences-metric-spaces
 open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-addition-real-numbers
+open import real-numbers.limits-sequences-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 ```
 
@@ -75,3 +78,20 @@ add-cauchy-sequence-ℝ {l1} {l2} u v =
       ( u)
       ( v))
 ```
+
+### If a sequence has a limit, there exists a modulus making it a Cauchy sequence
+
+```agda
+abstract
+  exists-cauchy-modulus-has-limit-sequence-ℝ :
+    {l : Level} (σ : sequence (ℝ l)) →
+    has-limit-sequence-ℝ σ →
+    is-inhabited (is-cauchy-sequence-ℝ σ)
+  exists-cauchy-modulus-has-limit-sequence-ℝ {l} σ (lim , is-lim) =
+    map-is-inhabited
+      ( is-cauchy-has-limit-modulus-sequence-Metric-Space
+        ( metric-space-ℝ l)
+        ( σ)
+        ( lim))
+      ( is-lim)
+```
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 9ca6388089..e21335fbd5 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -257,6 +257,16 @@ abstract
       ( leq-transpose-right-add-ℝ _ _ _ y≤z+x)
 ```
 
+### The difference of two real numbers is at most their distance
+
+```agda
+abstract
+  leq-diff-dist-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
+    leq-ℝ (x -ℝ y) (dist-ℝ x y)
+  leq-diff-dist-ℝ _ _ = leq-abs-ℝ _
+```
+
 ### Addition preserves distance between real numbers
 
 ```agda
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index 2aaf9e43a0..8b183862a6 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -44,6 +44,9 @@ is-limit-prop-sequence-ℝ {l} =
 
 is-limit-sequence-ℝ : {l : Level} → sequence (ℝ l) → ℝ l → UU l
 is-limit-sequence-ℝ {l} = is-limit-sequence-Metric-Space (metric-space-ℝ l)
+
+has-limit-sequence-ℝ : {l : Level} → sequence (ℝ l) → UU (lsuc l)
+has-limit-sequence-ℝ {l} = has-limit-sequence-Metric-Space (metric-space-ℝ l)
 ```
 
 ## Properties
diff --git a/src/real-numbers/sums-of-finite-sequences-of-real-numbers.lagda.md b/src/real-numbers/sums-of-finite-sequences-of-real-numbers.lagda.md
new file mode 100644
index 0000000000..1e26b21c1d
--- /dev/null
+++ b/src/real-numbers/sums-of-finite-sequences-of-real-numbers.lagda.md
@@ -0,0 +1,37 @@
+# Sums of finite sequences of real numbers
+
+```agda
+module real-numbers.sums-of-finite-sequences-of-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.natural-numbers
+
+open import foundation.universe-levels
+
+open import group-theory.sums-of-finite-sequences-of-elements-abelian-groups
+
+open import lists.finite-sequences
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.large-additive-group-of-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "sum operation" Disambiguation="of a finite sequence in ℝ" WD="sum" WDID=Q218005 Agda=sum-fin-sequence-ℝ}}
+extends the operation of [addition](real-numbers.addition-real-numbers.md) on
+the [real numbers](real-numbers.dedekind-real-numbers.md) to any
+[finite sequence](lists.finite-sequences.md) of real numbers.
+
+## Definition
+
+```agda
+sum-fin-sequence-ℝ : {l : Level} (n : ℕ) → fin-sequence (ℝ l) n → ℝ l
+sum-fin-sequence-ℝ {l} = sum-fin-sequence-type-Ab (ab-add-ℝ l)
+```