= Cesaro theorem for convergent sequences
{c}
{title2=$a_n\to a\ \Longrightarrow\ n^{-1}\sum_{k=1}^na_k\to a$}
If a <sequence> converges, its <Cesaro means> have the same limit. Given a small tolerance, split the average error into a fixed finite initial segment and a tail whose individual errors are below that tolerance. The first contribution is a fixed constant divided by $n$, while the second is bounded by the tolerance. The converse fails: the alternating <sequence> $(-1)^n$ has means tending to zero while its terms do not converge.
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