you should know that :

real number $\displaystyle a \in \mathbb{R}$ is limit of the sequence $\displaystyle x_n$ if for every $\displaystyle \varepsilon >0 $ there is natural number $\displaystyle n_0 \in \mathbb{N} $ such that for every natural number $\displaystyle n\ge n_0$ implies $\displaystyle |x_n-a|<\varepsilon$

$\displaystyle (\forall \varepsilon >0) (\exists n_0 \in \mathbb{N}) (\forall n\in \mathbb{N}) (n\ge n_0 \Rightarrow |x_n -a | < \varepsilon ) $

and that means

$\displaystyle \displaystyle \lim_{n\to \infty} x_n = a $

if that is true than sequence $\displaystyle x_n$ converges to $\displaystyle a $

or you can look at it like this

let the $\displaystyle (X,d)$ is metric space. for sequence $\displaystyle (x_n)_{n\in \mathbb{N}} \subset X$ we say that is Cauchy sequence if

$\displaystyle (\forall \varepsilon >0) (\exists n_0=n_0(\varepsilon) \in \mathbb{N}) (\forall n,m \in \mathbb{N}) (n,m\ge n_0 \Rightarrow d(x_n,x_m) < \varepsilon )$

meaning that, sequence is Cauchy sequence if

$\displaystyle \displaystyle \lim _{n,m \to \infty } d(x_n,x_m) = 0 $