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Two Limits Equal - Proof
Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means

Suppose $a_n \rightarrow c$ as $n \rightarrow \infty$. We want to show that $\dfrac{1}{n} \sum_{i=1}^n a_n \rightarrow c$ as $n \rightarrow \infty$.

I know I can use the definition to prove this by showing that for any given $\epsilon$, we can find an $N$ such that whenever $n \ge N$, we can have $|\dfrac{1}{n} \sum_{i=1}^n a_n-c| \le \epsilon$.

I am wondering whether there is another easier way to prove this.

Thank you very much. Hanna

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    [This](http://math.stackexchange.com/questions/210681/if-a-n-to-ell-then-hat-a-n-to-ell) should help.2012-12-04

2 Answers 2