Convergent sequence

A real sequence $(x_n)$ has limit of a sequence $L\in \mathbb{R}$ when for every $\epsilon >0$ there is $N$ such that $|x_n-L|<\epsilon$ for all $n\ge N$. Its limit of a sequence is unique. Extended-real convergence to $-\infty$ means that every real upper bound eventually exceeds all the terms; convergence to $+\infty$ is the corresponding lower-bound condition. The extended-real Fekete lemma is an example where the limit of a sequence need not be finite.

The Cesaro mean of the first $n$ terms of a sequence is their arithmetic average. If $a_n\to L$, then its Cesaro means also tend to $L$.

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.

For a positive sequence converging to $L\ge 0$, its cumulative geometric means converge to the same limit.

An increasing real sequence converges exactly when it is bounded above. For a bounded sequence, its limit is the supremum of its set of terms.