I need help finishing this proof using the Intermediate Value Theorem? Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$.
Here's what I have so far:
Let $h$ be a continuous function where $h = f -g$
Since $h(b)=f(b)-g(b)\leq 0$, and $h(a)=f(a)-g(a)\geq 0$, then $h(b)\leq 0 \leq h(a)$ is true.
So, by IVT there exists some $y$ such that...
And that's where I need help. I read what the IVT is, but I could use some help explaining how it applies here, and why it finishes the proof. Thank you!

There exists some $y \in (a,b)$ such that $h(y) = 0$. So $f(y) = g(y)$. The other case $a=b$ is trivial.
