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I am looking at Cesaro's average proof:

enter image description here

Please note that they started by saying that $\exists N_0$ such that for $n \geq N_0$: $|x_n-l| \leq \frac{\epsilon}{2}$

The first part in that picture they showed is bounded by $\frac{\epsilon}{2}$, so fine. The second part just comes from what I have said above. But there are $n-N_0+1$ terms in the second part and not $n-N_0$, no?

But even, if I am correct, then the proof is still correct, because $\frac{N_0}{n} > \frac{1}{n}$ and so $\frac{n-N_0+1}{n} < 1$.

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Yes, you are correct that the proof has a typo, and that it doesn't raise much of an issue.

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