Can anyone please tell me if my proof is fine?
Let N be a finite set. Then N = {n1,n2,....nk} has k elements. Let A be a subset of N and A does not equal N. Then A will either be empty or A will have less than k elements. If A is empty then A has no elements, consequently A is finite. If A has less than k elements, then k is finite so any number less than k is finite. Then this implies A is finite. Since A is any arbitrary subset of N, this implies all subsets of N will be finite.
Well it's correct but it would be a lot easier to say: Let $|N|=k$, and $A\subset N$, that is $|A|\leq |N|<k<\infty$. However, it depends on what you're allowed to use to prove the statement.