Induction involving binomial coefficients ${n \choose k}$

Question

Why, many times, in problems involving binomial coefficients, ${n \choose k}$, that we need induction to prove something, we use induction on $k$ instead on $n$. Since $n$ limits $k$, it seems strange to me to prove it for all $k$, since once proven, $k$ would "extrapolate" $n$, as it goes to infinity. You can say that $n$ also goes to infinity, but it sounds strange to me because when would $k$ then reach $n$?