Proving a formula with binomial coefficient when n=-1

Homework Statement

Prove that ##\binom{-1}{k}=(-1)^k##

The Attempt at a Solution

Using induction on ##k##,

##\binom{-1}{0}=1## which is true also for ##(-1)^0=1##

Assuming ##\binom{-1}{k}=(-1)^k##, then ##\binom{-1}{k+1}=(-1)^{k+1}##

Indeed when ##n=-1##, we can write rewrite this ##\frac{n!}{k!(n-k)!}## as ##\frac{(-1)^k(k)!}{k!}## to avoid negative factorial. Hence ##\binom{-1}{k+1}=\frac{(-1)^k(k+1)!}{(k+1)!}=(-1)^{k+1}##.

##\blacksquare##

I just want to confirm if my proof by induction method is valid.

Thank You