= Subsequential limit
{title2=$a_{n_k}\to L$}
A subsequential limit of a <sequence> $(a_n)$ is a value $L$ approached by some <subsequence> $(a_{n_k})$ with strictly increasing indices $n_k$. For $a_n=(-1)^n(1+1/n)$, the <subsequences> at <even integers> and <odd integers> give <subsequential limits> one and minus one. A finite value appearing among only finitely many initial terms does not become a <subsequential limit> merely by being a member of the <image of a function> represented by the <sequence>.
For a <bounded sequence> of <real numbers>, its <subsequential limits> form a nonempty <bounded set> that is also a <closed set>. The <Bolzano-Weierstrass theorem> guarantees nonemptiness, and its extreme values are the <limit inferior> and <limit superior>. The <sequence> in <sparse peaks can determine a limit superior> has the two <subsequential limits> zero and one, despite almost all indices in the <natural density> sense carrying zero.
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