If all subsequences $\{x_{n_i}\}$ of $\{x_n\}$ $ \lim_k\frac1k\sum_{i=1}^{k} {x_{n_i}}= y $ then $\lim_n x_n= y$

Question

Let $\{x_n\}_n$ be a real sequence and $y\in\mathbb{R}$ such that for all subsequences $\{x_{n_i}\}$ of $\{x_n\}$ we have $$ \lim_k\frac{1}{k}\sum_{i=1}^{k} {x_{n_i}}= y $$ My problem:

Why $\lim_n x_n= y$?

Hint: Use an elementary property of the Cesàro convergence.

Answer 1

This is trivial. Suppose $x_n$ does not tend to $y$. There is $\epsilon>0$ such that $|x_n-y|\ge\epsilon$ for infinitely many $n$. Wlog $x_n-y\ge\epsilon$ for infinitely many $n$; this gives a subsequence $x_{n_k}$ with $x_{n_k}\ge y+\epsilon$ for every $k$.