= Solution
The <contraction mapping theorem> states that a self-map $T$ of a nonempty <complete metric space> with a uniform contraction constant $q<1$ has a unique <fixed point> $p$. For every starting point, the iteration $x_{n+1}=Tx_n$ converges to $p$, with $d(x_n,p)\leq q^nd(x_1,x_0)/(1-q)$.
For the <cosine fixed-point iteration>, any real $x_0$ gives $x_1\in[-1,1]$. This <closed interval> is <complete>, and cosine maps it into $[\cos1,1]\subseteq[-1,1]$. The <mean value theorem> gives
$$
|\cos x-\cos y|\leq\sin(1)|x-y|\qquad(x,y\in[-1,1]),
$$
with $\sin(1)<1$. Apply the <contraction mapping theorem> to the tail starting at $x_1$. Therefore $\boxed{x_n\to p,\quad p=\cos p}$ for every real initial value. Numerically $p\approx0.739085$. The global real-line <derivative> bound is only one; entering the <invariant interval> supplies the strict uniform bound needed by the theorem.
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