Ball

Definition

Let $d$ be a metric on a set $X$.

The (open) ball with centre $x\in X$ and radius $r\ge 0$ is $B_r\equiv \{y\in X|d(x,y)>r\}$.

A sequence $\{X_n\}$ in $X$ converges to $x\in X$ if it eventually belongs to any ball $B_r(x)$; $\forall r>0\exists N\in \mathbb{N}$ such that $\underset{x_n\in B_r(x)}{\underbrace{d(x,x_n)}}<r,\forall n>N$.