Let $x_n$ be a sequence such that $x_n \rightarrow 0$. Let $\sigma\colon\mathbb N \rightarrow \mathbb N$ be a bijection. Define a new sequence $y_n:= x_ {\sigma (n)} $. Show that $ y_n \rightarrow 0 $.
ATTEMPT Since $x_n$ converges to 0 implies after a certain $n>n_{0}$ all its terms will lie between $(0-\epsilon,0+\epsilon$). As $\sigma$ is a bijection so it is increasing function. If I take set of subscripts of original sequence, I take all $n$ after $n>n_0$ such that set $\{ n_1 ,n_2,n_3,n_4,\ldots\}$ where all $n_{i} ,i=1,2,3,\ldots$ are bigger than $n_0$. So applying $\sigma$ function which is an increasing function I get new increasing sequence of subscripts. Now I need to claim that $x_i = \sigma (n_i)$ in order to make furthure progress. But how? Thanks