Theorem
Every convergent sequence is a Cauchy sequence.

Proof

Let \(\{x_n\}\) be a convergent sequence. Let \(\{x_n\} \rightarrow x\). This means that for any \(\epsilon > 0\), there must exist a number \(N \in \mathbb{N}\) such that

$$ \begin{align*} |x_n - x| < \frac{\epsilon}{2} \end{align*} $$

Hence if \(n, m > N\), then

$$ \begin{align*} |x_m - x_n| = |x_m - x + x - x_n| \leq |x_m - x| + |x - x_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align*} $$

From this we see that \(\{x_n\}\) is a Cauchy sequence as we wanted to show. \(\blacksquare\)

References