# Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence

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## Theorem

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

Let $x \in X$.

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

Then $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$ if and only if:

- $\norm {x_n - x} \to 0$

## Proof

From the definition of a convergent sequence in a normed vector space, we have that:

- $x_n$ converges to $x$

- for each $\epsilon > 0$ there exists $N \in \N$ such that $\norm {x_n - x} < \epsilon$.

From the definition of a convergent real sequence, we have that:

- $\norm {x_n - x} \to 0$

- for each $\epsilon > 0$ there exists $N \in \N$ such that $\size {\norm {x_n - x} - 0} < \epsilon$.

Since the norm is non-negative, we have that:

- $\norm {x_n - x} \to 0$

- for each $\epsilon > 0$ there exists $N \in \N$ such that $\norm {x_n - x} < \epsilon$.

We can therefore immediately deduce the result.

$\blacksquare$