Convergent Sequence is Cauchy Sequence/Normed Vector Space

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

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


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.