Alternating series test to prove the convergence of the series

I want to apply the alternating series test to prove the convergence of the series $\sum_{n\geq 1} (-1)^n u_n$ where $u_1=1$ and $\forall n\in \mathbb{N},\quad u_{n+1}=\frac{\cos(u_n)}{n^\alpha}$ where $\alpha>0$.

We can show that $0\leq u_n\leq 1$ for all $n$ and that $u_n\to 0$. Numerically, it seems that the sequence $u_n$ is decreasing after a certain point, but I don't see how to prove it.