Limit of sequence of real numbers Let $\{x_n\}$ be sequence of real numbers such that $\lim_{n\to\infty} x_n=p$. 
Also if $m\to \infty$ then $n\to \infty$ and converse.
How to prove strictly $\lim_{m\to\infty} x_n=\lim_{n\to\infty} x_m=p$?
I met this feint in many books and also in my last problem that I solved but I can't understand this.

Well let's go back to the definition of limits as other answers seems to call the result which you willing to demonstrate :
Def of $(x_{n})$ converges to $p$ L : $\forall \varepsilon > 0, \exists N \in \mathbb N, \forall n>N,  x_{n}\in B(p,\varepsilon) $
For $\varepsilon > 0$, as $(x_{n})$ converges to $p$ let N be such an integer, as $m(n) \rightarrow \infty $ , let $M$ be an integer such that $  \forall n>M, m(n)>N $, and we have $\forall n>M,  x_{m(n)}\in B(p,\varepsilon) $
