TITLE: Verifying Proof of decreasing functions and function composition increasing.
QUESTION [0 upvotes]: I want to prove the following: 

If $f$ and $g$ are decreasing functions on $I$ and $f \circ g$ is defined on $I$, then $f \circ g$ is increasing on $I$.

My attempt: Let $x_1,x_2$ be in $I$ such that $x_1\lt x_2$ and let $f$ and $g$ be decreasing. Then $g(x_1)\gt g(x_2)$. 
Then $f \circ g (x_1)= f(g(x_1))$ and $f \circ g(x_2)=f(g(x_2))$
$f(g(x_1)) < f(g(x_2))$
So, $f \circ g$ is increasing on I.

REPLY [0 votes]: If $x_2 > x_1 \implies g(x_2) < g(x_1) \implies f(g(x_2)) > f(g(x_1)) \implies (f\circ g)(x_2) > (f\circ g)(x_1) \implies f\circ g $ is an increasing function.
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