$\displaystyle $$\text{Let }\left( \text{X},\text{ d} \right)\text{ a metric space}.\text{ Show that }{{\left\{ {{\text{x}}_{\text{n}}} \right\}}_{\text{n}\in \mathbb{N}}}~\text{satisfies }{{\text{x}}_{\text{n}}}\to \text{x when n}\to \infty \text{ if and only if every subsequence }\!\!\{\!\!\text{ }{{\text{x}}_{{{n}_{k}}}}{{\text{ }\!\!\}\!\!\text{ }}_{k\in \mathbb{N}}}\subset {{\left\{ {{\text{x}}_{\text{n}}} \right\}}_{\text{n}\in \mathbb{N}}}\text{has a sub}-\text{subsequence}~{{\text{ }\!\!\{\!\!\text{ }{{\text{x}}_{{{n}_{{{k}_{j}}}}}}\text{ }\!\!\}\!\!\text{ }}_{j\in \mathbb{N}}}\subset {{\text{ }\!\!\{\!\!\text{ }{{\text{x}}_{{{n}_{k}}}}\text{ }\!\!\}\!\!\text{ }}_{k\in \mathbb{N}}}\text{ such that }{{\text{x}}_{{{n}_{{{k}_{j}}}}}}\to x\text{ when j}\to \infty $$$