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Let $\lbrace a_n\rbrace$ be a convergent sequence with $a_n \to L$

Define $$ b_n = \frac{ a_1 + a_2 + ... a_n}{n} \forall n \in \mathbb Z_+ $$

Prove that $\lbrace b_n\rbrace$ is convergent with $ b_n \to L$

I tried to go through $\displaystyle \lim_{n \to \infty} f \{a_n \} = f \lim_ {n \to \infty} \{ a_n \} $ but I can't figure out $f$.

Semiclassical
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S.Dan
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1 Answers1

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Recall the definition of convergence: $\forall \epsilon \exists N$ such that $\forall n > N$, $|a_n-L| < \epsilon$.

As a hint, try it with $L=0$.

Another hint: for a given $\epsilon$, you want the average of the first $k$ $a_n$s, but you can use as many $|a_n-L|<\frac \epsilon 2$ as you need to get the average low!

Clinton Bradford
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