Take the sequence of cosine iterates as follows: $x_0 = π/4$, $x_{n+1} = \cos(x_n)$ for n = 0, 1, 2, . . . .
One can prove that this sequence converges, say to p. Show that (p, cos p) is the point where the graph of y = cos x intersects the line y = x.
Any hints or pointers?
Suppose that $x_n$ converges to $p$. By the continuity of $\cos(x)$, we may state $$ \cos p = \cos\left(\lim_{n \to \infty}x_n\right) = \lim_{n \to \infty}\cos x_n = \lim_{n \to \infty} x_{n+1} = \lim_{n \to \infty} x_n = p $$