I'm trying to prove that if $\{x_n\}\to 0$ and $y_n=\frac{x_1-x_2-...-x_n}{n}$ then $\{y_n\}\to 0$. One can easily show that $\left| y_n\right|<x_{max}$, but this doesn't seem to tell us much.
I'd appreciate any hints on how to prove the above.
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Well $y_n = \frac{x_1}{n} - \sum \limits_{i=2}^n \frac{x_i}{n}$, which is asymptotically just $-\sum \limits_{i=2}^n \frac{x_i}{n}$. Now you can solve it or Google the solution pretty quickly. – Matias Heikkilä Sep 30 '16 at 09:39
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Didn't find anywhere what how to sum $\frac{x_i}{n}$. Can you please elaborate? We don't know what $x_i$ is for any $i$. – sequence Sep 30 '16 at 09:51