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.

Let $m(n)$ be a monotone function $\mathbb N\to\mathbb N$, then $(x_m):=(x_{m(n)})$ is a subsequence of $(x_n)$. Because each subsequence of a convergent sequence converges to the same limit as the original sequence, you have $\lim_{n\to\infty} x_{m(n)} = p$.
