Bernoulli inequality for integer powers

Show that for every real $x>-1$ and every integer $n\ge2$, $(1+x)^n\ge1+nx$.

Induction. For $n=2$, $(1+x)^2=1+2x+x^2\ge1+2x$. If it holds for $n$, multiply by $1+x>0$: $(1+x)^{n+1}\ge(1+nx)(1+x)=1+(n+1)x+nx^2\ge1+(n+1)x$.