# Convergent Sequence is Cauchy Sequence/Normed Vector Space

## Theorem

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

Every convergent sequence in $X$ is a Cauchy sequence.

## Proof

Let $\sequence {x_n}$ be a sequence in $X$ that converges to the limit $L \in X$.

Let $\epsilon > 0$.

Then also $\dfrac \epsilon 2 > 0$.

Because $\sequence {x_n}$ converges to $L$, we have:

$\exists N: \forall n > N: \norm {x_n - L} < \dfrac \epsilon 2$

So if $m > N$ and $n > N$, then:

 $\ds \norm {x_n - x_m}$ $=$ $\ds \norm {x_n - L + L - x_m}$ $\ds$ $\le$ $\ds \norm {x_n - L} + \norm {x_m - L}$ Triangle Inequality Axiom of Norm $\ds$ $<$ $\ds \frac \epsilon 2 + \frac \epsilon 2$ (by choice of $N$) $\ds$ $=$ $\ds \epsilon$

Thus $\sequence {x_n}$ is a Cauchy sequence.

$\blacksquare$