Proving problem about relative extrema

Question

Prove that if $f$ is increasing on $[a, b]$ and $g$ is increasing on $[f(a),f(b)]$ then if $g \circ f$ exists on $[a,b]$, $g \circ f$ is increasing on $[a,b]$.

It's about the relative extrema of functions, and I don't know how to do this.

Thanks for the answers!

Answer 1

Take some $x,y\in[a,b]$ such that $x\le y$. Then $a\le x\le y\le b$, and since $f$ is increasing on $[a,b]$ we have that $f(a)\le f(x)\le f(y)\le f(b)$, so $f(x),f(y)\in[f(a),f(b)]$ and $f(x)\le f(y)$. Therefore, since $g$ is increasing on $[f(a),f(b)]$ (try to figure it out first!):

$g(f(x))\le g(f(y))$, which is the same as $(g\circ f)(x)\le (g\circ f)(y)$. So we took $x,y\in[a,b]$ such that $x\le y$ and we got that then $(g\circ f)(x)\le (g\circ f)(y)$, hence we can conclude

$g\circ f$ is increasing on $[a,b]$