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Suppose we have bounded, monotonically decreasing $\epsilon_i$, such that as $i \rightarrow \infty, \epsilon_i \rightarrow 0$.

How can I show that $\lim_{n \rightarrow \infty} \frac{\sum_{i=1}^\infty \epsilon_i}{n} = 0$? Is this even necessarily true? Tried Kronecker's lemma, as well as comparison tests to no avail. Seems like a pretty fundamental fact to prove.

ringwraith10
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  • You don't need the monotone property. Proved three ways in the link above. – RRL Oct 15 '17 at 08:10
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    Don't you mean $\lim_{n \to \infty} \frac{\sum_{i=1}^n \epsilon_i }{n} = 0$. Otherwise it is trivial if the series is convergent - which is not always true given your hypotheses. – RRL Oct 15 '17 at 08:21

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