Limit of an integral sequence

Question

Let $(I_n)_{n\ge1}$ be a sequence, $$I_n=\frac{1}{n} \int_{\frac{1}{n}}^1 \ln(1+\cos(x)) \, dx, \forall n\ge1$$ Show that $(I_n)_{n\ge1}$ converges and find its limit. It is clear that $I_n>0$ but how do I prove that it is decreasing? As for the limit, I suppose it is $0$ and the squeeze theorem could be used to show that.

Answer 1

Define

$$a_n = \frac1n$$ $$b_n = \int_{\frac1n}^1 \log(1 + \cos x) dx$$

Now,

$$a_n \to 0$$ $$b_n \to \int_0^1 \log(1 + \cos x) dx \in \mathbb{R}$$

Therefore $a_n b_n \to 0$.