# Why does $\lim_{n\to\infty} z_n=A$ imply $\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A$?

I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following.

Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A.$$

Is there a clever way to write the limit to make it more approachable? Thank you.

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It'll also be helpful, math.columbia.edu/~nironi/stolz-cesaro.pdf – Ehsan M. Kermani Jan 30 '12 at 4:11
Thanks for the link, @ehsanmo. – Dedede Jan 30 '12 at 4:19

Let $\epsilon>0$. Since $\lim_{n\to\infty}z_n=A$, there exists an positive integer $N$ such that $$\tag{1}|z_n-A|<\frac{\epsilon}{2}\mbox{ whenever }n\geq N.$$ For this fixed $N$, we can find another positive integer $N'$ such that $$\tag{2}\sum_{k=1}^{N-1}|z_i-A|\leq \frac{N'\epsilon}{2}.$$

Hence, if $m\geq\max\{N, N'\}$, then $$\left|\frac{1}{m}(z_1+\cdots+z_m)-A\right|=\left|\frac{1}{m}\sum_{k=1}^{m}(z_i-A)\right|$$ $$\leq\left|\frac{1}{m}\sum_{k=1}^{N-1}(z_i-A)\right|+\left|\frac{1}{m}\sum_{k=N}^{m}(z_i-A)\right|$$ $$\leq\frac{1}{m}\sum_{k=1}^{N-1}|z_i-A|+\frac{1}{m}\sum_{k=N}^{m}|z_i-A|$$ $$\leq \frac{\epsilon}{2}+\frac{1}{m}(m-N+1)\frac{\epsilon}{2}\leq \epsilon$$ where we have used $(1)$ and $(2)$ in the last inequality. This proves that $$\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A.$$

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Thank you Paul, I find this answer very clear and easy to understand. – Dedede Jan 30 '12 at 2:50
@Dedede: You are welcome. – Paul Jan 30 '12 at 3:45

We have that

$$\frac{z_1+\cdots+z_n}{n}-A=\frac{(z_1-A)+\cdots+(z_n-A)}{n},$$

so it suffices to prove the statement when $A=0$. The $\epsilon$-$N$ definition of the limit tells us that for any positive $\epsilon>0$, there is an $N$ such that $n>N\implies |z_n|<\epsilon$, which subsequently implies

$$\left|\frac{z_1+\cdots+z_{k+N}}{k+N}\right|=\left|\frac{z_1+\cdots+z_N}{k+N}\,+\,\frac{z_{N+1}+\cdots+z_{N+k}}{k+N}\right|$$

$$\le \left|\frac{z_1+\cdots+z_N}{k+N}\right|\,+\,\left|\frac{z_{N+1}+\cdots+z_{N+k}}{k+N}\right|\le \left|\frac{z_1+\cdots+z_N}{k+N}\right|+\epsilon\stackrel{k\to\infty}{\longrightarrow}\epsilon.$$

(We also used the triangle inequality above.) Now take $\epsilon\to0^+$ and use the squeeze theorem.

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Thanks again for your help, anon. – Dedede Jan 30 '12 at 2:50