generated from smyalygames/quartz
6 lines
409 B
Markdown
6 lines
409 B
Markdown
# Definition
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Let $d$ be a [[Metric|metric]] on a set $X$.
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The (open) **ball** with centre $x \in X$ and radius $r \geq 0$ is $B_{r} \equiv \{ y \in X | d(x,y) \gt r\}$.
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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 $\underbrace{d(x, x_{n})}_{x_{n} \in B_{r}(x)} \lt r, \; \forall n \gt N$. |