When proving $\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}$ per induction over $n$, why don’t we need induction over $r$?

Question

Proving this formula by induction is done by induction over $n$: $$ \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1} $$ My question is: Why don't we need to show this seperately for $r \to r+1$ too?

Answer 1

Since $r$ is depending on $n$ and limited by it, so when you take care of $n$ you have automatically have taken care of $r$ as well. Thus we do not need another induction for $r$ or $r+1$.