Let $\displaystyle x\in [0,1]$ and $\displaystyle n$ be any positive integer, prove $\displaystyle (1-x^2)^n\geq1-nx^2$.

It can be easily proved by induction on $\displaystyle n$, but I would like to use binomial theorem to prove it. By binomial theorem, $\displaystyle (1-x^2)^n=1-nx^2+...$. The proof is completed if the remaining "..." part in the above expansion can be proved to be $\displaystyle \geq0$, but I have no idea how to prove it. Could anyone help me with this problem? Thanks!