combinatorics - Alternative Proof for ${nchoose k}$ is integer

I have seen different type of induction proofs on this case, but trying an alternative approach I tried Induction to show that ${n\choose k}$ in binomial coefficient is an integer, where both n and k are non-negative integers.

Base case: For k = 0, ${n\choose 0}$ = 1, and is integer.

Inductive Hypothesis: For k= n-1, Assume ${n\choose n-1}$ is integer. (That's not even assumption but a fact, in fact.)

Finally, induction: For k = n, ${n\choose n}$ is integer because it's 1.

Is this a proof? Is this a thing? What is it?