Definition

Let d be a Metric on a set X.

The (open) ball with centre x \in X and radius r \geq 0 is B_{r} \equiv \{ y \in X | d(x,y) \gt r\}.

A sequence \{ X_{n} \} in X converges to x \in X if it eventually belongs to any ball B_{r}(x); \forall r \gt 0 \; \exists N \in \mathbb{N} such that \underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N.