= Peak of a sequence
{title2=$a_k\geq a_m\quad\text{for every }m>k$}
= Peaks of a sequence
{synonym}
A peak of a <sequence> of <real numbers> is an index $k$ whose value is at least every later value. This is a condition on the entire tail, stronger than comparing only adjacent terms. Equality is allowed: every index of a constant <sequence> is a <peak of a sequence>.
Infinitely many <peaks of a sequence>, listed in increasing order of index, give a nonincreasing <monotone subsequence>. If only finitely many exist, every index beyond the last has a larger later term; repeatedly choosing one gives a strictly increasing <monotone subsequence>. This proves the <monotone subsequence theorem>.
A <peak of a sequence> of $(-a_n)$ supplies a lower bound for all subsequent terms of $(a_n)$. If both $(a_n)$ and $(-a_n)$ have a <peak of a sequence>, the two selected values bound a common tail. Adding bounds for the finite initial segment proves that $(a_n)$ is a <bounded sequence>.
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