Proving that an $n$-element heap has at most $\lceil \frac{n}{2^{h+1}-1} \rceil$ nodes I am having trouble proving that an n-element heap can have at most $\lceil \frac{n}{2^{h+1}-1} \rceil$ nodes.

Please note that I am proving a loose bound.

First I proved that a complete binary tree can have at most $2^{h+1} - 1$ nodes. However, I don't know how to use this fact to inductively prove the formula above?

The only thing thus far in my inductive proof is showing the base case ($h=0$), which is trivial.
Probably, you mean this:
A heap of size $n$ has at most $\lceil \frac{n}{2^{h+1}} \rceil$ nodes with height $h$.

Proof can be found for example here:
http://www.cs.sfu.ca/CourseCentral/307/petra/2009/SLN_2.pdf