Your proof is not valid, assuming I understood you correctly.
This is the attempt given:
The problem is you started out by assuming $a + b = 88$. You just assumed what you wanted to prove!
Here is the question again:
Can you prove this:
if $a,b$ are positive integers,
and if $a + b + ab = 2020$,
Notice that the fact that $a, b$ are positive integers is very necessary here. Otherwise, you could pick any rational $a,b$ with $(a + 1)(b + 1) = 2021$ (in this case $a$ can be any rational number) and you would have a solution. In general $a + b \ne 88$ if $a,b$ are rational.
Thus your proof should be suspicious: you haven't used any properties about integers, as far as I can see. Your proof would also conclude that $a + b = 88$ if $a,b$ are rational, and this is not a true result! Therefore, your proof cannot be valid.
The correct proof is as leticia gives: write $(a + 1)(b+1) = 2021 = 47 \cdot 43$, and use prime factorization -- a property of positive integers -- to derive your result.