how can I prove that $\frac{\arctan x}{x }< 1$? I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.

Hint
Try the function $f(x)=\arctan{x} - x$. It's derivative is $\frac{-x^2}{1+x^2}$. Now use the monotonicity to get the required inequality.
