I am having a little trouble understanding this part of a proof.
There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$
Now, $\text{J}_0 = \frac{\pi ^3}{24} $
The part of the proof I am unable to understand says that $x \lt \frac{\pi}{2}\sin x $ for every $0 \lt x \lt \frac{\pi}{2}$
They are saying that thus inequality implies that $$\text{J}_n \lt \frac{\pi^2}{4} \int_0^{\frac{\pi}{2}}\sin^2x \cos^{2n} x dx $$
I am unable to supply the intermediate steps. Please help.