Convergence of a Sequence in terms of Open Sets

Statements

Definition. A sequence $\{x_i{\}}_{i\in \mathbb{N}}$ is said to converge to $p$ if, for every neighborhood $U$ of $p$, there exists a positive integer $N$ such that for all $i\ge N,x_i\in U$. In this case $p$ is said to be a limit of the sequence $\{x_i{\}}_{i\in \mathbb{N}}$ and denoted as $x_i\to p$ or ${\lim }_{i\to \infty }{x}_i=p$.

References

@Tu11 Definition A.54