$a_1 =π/4$, $a_{n+1}= \cos(a_n)$. Prove that $\lim(a_n) =c$ when $c =\cos(c)$

Question

I have been struggling for too long, Does anyone have an idea how to solve this one ? $a_1 =π/4\:$ and $a_{n+1}=\cos(a_n)$

Prove that $$\lim_{a_n \to \infty} =c\;\text{ when } \; c = \cos(c)$$

Answer 1

If you have $$a_{n+1}=\cos(a_n)$$ If the limit $L$ exists, it is, by definition, the solution of $$L=\cos(L)$$ and this is Dottie number.

The value of $a_1$ has nothing to do.

So, you need to prove the convergence of the series.