Cardinality of strictly increasing natural-number sequences ($|\{a\in {\mathbb{N}}^{\mathbb{N}}:a_n<a_{n+1}\}|=2^{{\aleph }_0}$)

Strictly increasing natural-number sequences form a subset of ${\mathbb{N}}^{\mathbb{N}}$, giving the continuum upper bound. Map each binary sequence $ϵ$ to $a_n=2n+{ϵ}_n$ for the matching lower bound: adjacent values differ by at least one and the binary sequence can be recovered.