Define the sequence $a_n$ recursively by
$a_{n+1}=a_n+\cos a_n$ for all $n\in \mathbb N$ with $a_1=1$
I've just begun with introduction to analysis and have witnessed these two sequences from the text. However, from the knowledge I've learned so far, I don't know how to prove that these sequeces converge and to what values they converge. How can I prove the convergence of these sequences?
Taking $f(x) = x + \cos(x)$, we've defined $$ a_{n+1} = f(a_{n}) $$ Note that $f'(x) = 1 - \sin(x)$, so that $|f'(x)| \leq 1/2$ for $x \in (\pi/6,5\pi/6)$, which contains $a_0 = 1$. Moreover, $f(x) \in (\pi/6,5\pi/6)$ for every $x \in (\pi/6,5\pi/6)$.
This means that $f$ defines a contractive mapping on $(\pi/6,5\pi/6)$; that is, $|f(x)- f(y)| \leq \frac 12 |x-y|$. Conclude that the sequence as defined must converge to a fixed point of $f$; that is, $a_n \to a$, and $a$ is the unique element of $(\pi/6,5\pi/6)$ satisfying $f(a) = a$.
See the Banach fixed-point theorem.