Convergent sequences are Cauchy

Statement

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

Remarks

Let $L$ be the limit and let $\varepsilon > 0$. Take $N$ with $|a_n - L| < \varepsilon/2$ for $n \ge N$. Then $|a_m - a_n| \le |a_m - L| + |L - a_n| < \varepsilon$ for $m, n \ge N$. The statement holds in each metric space. The converse holds in $\mathbb{R}$, but it fails in $\mathbb{Q}$ and in other spaces that are not complete.

Explanations

L1 human

If all late terms are close to one number $L$, then they are also close to each other. Two terms that are each within $\varepsilon/2$ of $L$ are within $\varepsilon$ of each other.