# Category:Convergent Sequences (Normed Vector Spaces)

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This category contains results about **Convergent Sequences** in the context of **Normed Vector Spaces**.

Definitions specific to this category can be found in Definitions/Convergent Sequences (Normed Vector Spaces).

Let $\struct {X, \norm {\,\cdot \,} }$ be a normed vector space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

Let $L \in X$.

The sequence $\sequence {x_n}_{n \mathop \in \N}$ **converges to the limit $L \in X$** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Convergent Sequences (Normed Vector Spaces)"

The following 15 pages are in this category, out of 15 total.

### C

- Constant Sequence in Normed Vector Space Converges
- Convergence in Direct Product Norm
- Convergence of Product of Convergent Scalar Sequence and Convergent Vector Sequence in Normed Vector Space
- Convergent Sequence in Normed Vector Space has Unique Limit
- Convergent Sequence in Normed Vector Space is Bounded
- Convergent Sequence in Normed Vector Space is Weakly Convergent
- Convergent Sequence is Cauchy Sequence/Normed Vector Space
- Convergent Sequences in Vector Spaces with Equivalent Norms Coincide
- Convergent Subsequence of Cauchy Sequence/Normed Vector Space