Given a sequence $A=(a_1,\ a_2,\ \ldots)$ one can define the records of $A$ as numbers $a_n:n\in\mathbb{Z}^+$ such that $a_n>a_m$ whenever $n>m.$ So you start at 1 and write down every number larger than all preceding numbers. (Of course you could also look for record-small numbers in just the same way.)
Is there a standard term for the terms $a_n:n\in\mathbb{Z}^+$ such that $a_n
These are somewhat more tricky to work with since you can't prove membership by checking finitely many values. But they are often defined and useful. Surely there is a standard term for this somewhere; I'm loathe to invent terminology except when absolutely necessary.