Theorem
Cauchy sequences are bounded.

Proof

Let \((a_n)\) be a Cauchy sequence and let \(\epsilon = 1\). By definition, this means that there exists a number \(N \in \mathbb{N}\) such that when \(n,m \geq N\)

$$ \begin{align*} |a_n - a_m| < 1 \end{align*} $$

Fix \(a_n\) such that \(n = N+1\). Since \(N+1 > N\), then the following holds for all \(m \geq N\)

$$ \begin{align*} |a_m - a_{N+1}| < 1 \end{align*} $$

By the reverse triangle inequality, then

$$ \begin{align*} |a_m| - |a_{N+1}| \leq |a_m - a_{N+1}| < 1 \end{align*} $$

Hence,

$$ \begin{align*} |a_m| < 1 + |a_{N+1}| \end{align*} $$

From this, we can see that we have established a bound on all the terms of the sequence starting after the \(N\)th term. Now, define

$$ \begin{align*} M = \max\{|a_1|, |a_2|, |a_3|, ..., |a_N|, |a_{N+1}|+1\}, \\ \end{align*} $$

Hence, \(|a_n| \leq M\) for all \(n \in \mathbb{N}\). \(\blacksquare\)

References