TITLE: On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $
QUESTION [38 upvotes]: I have this problem I'm working on.
Hints are much appreciated (I don't want complete proof):
In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $ 
I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.
Thanks!

REPLY [2 votes]: WLOG, the $x_n$ converge to $0$ (otherwise consider the differences $x_n-x$), and stay confined in an $\epsilon$-neighborhood of $0$ after $N_\epsilon$ terms.
Then the average of the first $m$ terms is bounded by 
$$\frac{N\overline{x_N}+(m-N)\epsilon}m,$$ which converges to $\epsilon$. So you can make the average as close to $0$ as you like.