$I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n \ dt$ converges to 0? How one can prove that the sequence $\left ( I_n \right )$ defined as $$ I_n = \int_{0}^{\frac{\pi}{2}}(\cos t)^n  \ dt, $$ $n \in \{ 0,1,2,...\}$ converges to $0$?
Is easy to show, by the way, that the sequence is decreasing because, for $t  \in  (0, \pi/2)$, $$(\cos t)^{n+1}<(\cos t)^n \Rightarrow I_{n+1} < I_{n}, \ \forall n $$

With integation by parts one may show that
$$I_{n}=\dfrac{n-1}{n}I_{n-2}$$
then from $I_0=\dfrac{\pi}{2}$ and $I_1=1$, both odd and even terms go $0$ as $n\to\infty$
