$a_{n+1} = \cos(a_n)$ with $a_1 = x$ Let  $(a_n)_{n=1}^{\infty}$ be a sequence defined by 
$a_1=x$  and $  a_{n+1}=\cos(a_n)$ then check the  the convergence of sequence .
Here what Approached is like 
$-1\leq \cos x \leq +1$ so we get $a_2 = cos(\mathbb{I_1})$ in general $a_{n+1} = cos(\mathbb{I_n})$ and cos being an even function $\mathbb{I_2}$ onwards  it will contain elements $0 \leq x\leq c  $ and  $c<1$  hence it converges to $cos (0)=1$
It that a fair argument
 A: For the purposes of this answer, let superscript denote compositions. You want to evaluate:
$$y=\lim_{n \to \infty} \cos^n{(x)}$$
$$y=\lim_{n \to \infty} \cos{\left(\cos^{n-1}{(x)}\right)}$$
$n-1$ goes to infinity the same as $n$
$$y=\lim_{n \to \infty} \cos^{n-1}{(x)}$$
$$y=\cos{y}$$
The unique solution to this equation is called the Dottie number $=D \approx 0.739085$. Now to prove convergence for all initial real $a_0$. After one composition, $-1 \leq a_n \leq 1$. Now that the sequence is bounded, we want to prove that it is either monotonically increasing or decreasing. If it is, it must converge to something within the interval. This cannot be done for the sequence as it is because it alternates about $D$. The operation must be changed from $\left (a_{n+1}=\cos{a_n}\right)$ to $\left(a_{n+1}=\cos{\left(\cos{a_n}\right)}\right)$
We want to prove that:
$$a_n \leq a_{n+1} \leq D \quad \text{if} \quad a_n \leq D$$
$$a_n \geq a_{n+1} \geq D \quad \text{if} \quad a_n \geq D$$
$$\text{Define } f(x)=\cos{\left(\cos{x}\right)}-x \quad\quad f(D)=0$$
Since $D$ is the only zero of $f(x)$, $f(x)$ is continuous, and the end behaviors $\infty, -\infty$:
$$f(x) \geq 0 \text{ when } x \leq D$$
$$f(x) \leq 0 \text{ when } x \geq D$$
The relationship between adjacent terms in the sequence is verified. Next, we must show that the sequence stays on the same side of $D$. Observe that after two compositions, $a_n$ is bounded: $0 \leq \cos{1} \leq a_n \leq 1 \quad$ since $\quad 1 \leq \frac{\pi}{2}$
$$\text{Define } f(x)=\cos{\left(\cos{x}\right)}-D \quad\quad f(D)=0$$
$$f'(x)=\sin{x}\sin{\left(\cos{x}\right)}$$
$$f'(D)=\sin^2{D}$$
Since the derivative is positive at $D$, $f(x)$ is continuous, and $D$ is the only zero of $f(x)$ in the interval $[0, 1]$:
$$f(x) \geq 0 \text{ when } D \leq x \leq 1$$
$$f(x) \leq 0 \text{ when } 0 \leq x \leq D$$
Therefore, the two original inequalities are proven, and the sequence converges to $D$.
