Convergent sequence is Cauchy

Convergent sequence is Cauchy: Let $(X,d)$ be a metric space and let $(x_n)$ be a convergent sequence with $x_n\to x$ in $X$. Then $(x_n)$ is a Cauchy sequence ; that is, for every $\varepsilon>0$ there exists $N$ such that if $m,n\ge N$ then $d(x_m,x_n)<\varepsilon$.

Together with the converse implication in complete metric spaces , this links limits to intrinsic “eventual closeness” of sequence terms.