tags:definition

Limit of a Sequence

Context

Statements

Definition (@Men90, Definition 5.1, @Eck23, Definition 3.1). Let $(X,d)$ be a metric space , $\{x_i{\}}_{i\in \mathbb{N}}\subset X$ be a sequence. A point $x$ is said to the limit of the sequence $\{x_i{\}}_{i\in \mathbb{N}}$ if, for every $\epsilon >0$, there exists a positive integer $N>0$ such that, for every $n>N$, the inequality $d(x,x_n)<\epsilon$ holds. It is also said that the sequence $\{x_i{\}}_{i\in \mathbb{N}}$ converges to $x$.

Remark

A sequence is convergent if the terms of the sequence are getting closer and closer to some point $x\in X$.