I am having difficulty answering the following question on "Notes on Set Theory" by Moschovakis, 1st edition:
Prove that the set $K$ of all monotone real functions on the closed interval $[0,1]$ have the same cardinality with $\mathbb R$.
We can inject $\mathbb R$ into $K$ by sending any real number to its constant function, e.g. $5/2$ gets mapped to the constant function $5/2$ on $[0,1]$. By Schroder-Berstein theorem, it suffices to provide an injection from $K$ to $\mathbb R$. As the set of constant functions in $K$ have cardinality of $\mathbb R$, strictly decreasing function in $K$ (by which I mean that such a function $f$ has some $a > b$ st. $f(a) <f(b)$, i.e. $f$ need not be strictly decreasing in the whole interval) is in bijective correspondence with strictly increasing function in $K$, and $|\mathbb R^3| = |\mathbb R|$, it suffices to show that the set of strictly increasing function in $K$ injects to $\mathbb R$. However, I am not sure how I can provide such an injection.