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**topsquark** Suppose a series $\displaystyle z_n \to \alpha$ as $\displaystyle n \to \infty$.

(That is to say $\displaystyle ( z_n )$ has a limit point $\displaystyle \alpha$ at infinity.)

Prove that

$\displaystyle \lim_{n \to \infty} \frac{z_1 + z_2 + ~...~ + z_n}{n} = \alpha$

I have found a number of series where I can prove it case by case, but am having some trouble generalizing. In addition, if the series $\displaystyle z_n = 1 - \frac{1}{n}$ I am having an embarrassing time showing the result at all.

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