Cardinality of strictly increasing natural-number sequences
= Cardinality of strictly increasing natural-number sequences
{title2=$|\{a\in\mathbb N^{\mathbb N}:a_n<a_{n+1}\}|=2^{\aleph_0}$}
Strictly increasing natural-number <sequences> form a <subset> of $\mathbb N^{\mathbb N}$, giving the continuum <upper bound>. Map each binary <sequence> $\epsilon$ to $a_n=2n+\epsilon_n$ for the matching <lower bound>: adjacent values differ by at least one and the binary <sequence> can be recovered.