Prove inequality $ \frac{x^2}{(1 + x^2)^n}\ < \frac{1}{n}\ $ for any x [closed]

I'm trying to prove next inequality: $ \frac{x^2}{(1 + x^2)^n}\ < \frac{1}{n}\ $for $|x| < 1$ and any $n \in\ N$

I've tried to use induction (can't make induction step) and taylors formula (I got result depends on x).