Cauchy sequences are bounded

Statement

Let $(a_n)$ be a Cauchy sequence of real numbers. Then $(a_n)$ is bounded.

Remarks

Take $\varepsilon = 1$ to get $N$ with $|a_n - a_N| < 1$ for all $n \ge N$. Then $|a_n| \le \max(|a_0|, \dots, |a_{N-1}|, |a_N| + 1)$ for all $n$. The statement holds in each metric space. The proof of the Cauchy criterion uses this fact.

Explanations

L1 human

From some index on, all terms are within distance $1$ of one fixed term. The finitely many terms before that index are bounded too, so the whole sequence is bounded.