Convergence of a Sequence in terms of a Metric

Statements

Definition. Let $(X,d)$ be a metric space. Let $\{a_i{\}}_{i\in \mathbb{N}}\subset X$ be a sequence. A point $a\in X$ is said to be the limit of the sequence $\{a_i{\}}_{i\in \mathbb{N}}$ if ${\lim }_{n\to \infty }d(a,a_n)=0.$ In this case, the sequence is said to converge to $a$ and this is denoted as ${\lim }_{n\to \infty }{a}_n=a$.

Note that the Induced Topology of a Metric implies that the convergence of a sequence can also be stated in terms of open sets.

References

@Men90 Definition 5.2