[Math] If $f$ and $g$ are monotone increasing and if the composite function $f\circ g$ is defined, then it is also monotone increasing.

Question

For this question the definition for monotone increasing is that $f$ is monotone increasing if for all $x_1<x_2$ where $x_1,x_2\in I$, $f(x_1) \leqslant f(x_2)$.

I have to apply that definition to $f$, $g$ and $f\circ g$ but I'm not quite sure exactly how.

Answer 1

Hint:

$$x\le y\to g(x)\le g(y)\to f(g(x))\le f(g(y))$$