Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $(x_k)_{k\in\mathbb{N}}$ be a sequence with $x_k\in X$ for every $k\in\mathbb{N}$, and let $x\in X$. Then $x_k\to x$ in $(X,d)$ if and only if for every $r>0$ there exists $N\in\mathbb{N}$ such that
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\begin{align*}
x_k\in B(x,r)
\end{align*}
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for every $k\in\mathbb{N}$ with $k\ge N$, where $B(x,r)=\{y\in X:d(y,x)<r\}$ denotes the open ball of radius $r$ centered at $x$.