Is there a limit of $\cos(n!)$? I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks
I suspect that you have miscopied the problem, or your source has an error. I will argue that, while your statement is extremely likely to be true, we probably do not know enough to prove it.

Here is why. Suppose, hypothetically, that 
$$\frac{1}{2 \pi} = \sum_{i \geq 0} \frac{d_i}{i!}$$
where all the $d_i$ are integers and, for $i$ sufficiently large, each $d_i$ is either $0$ or $1$. This is extraordinarily unlikely. However, if we knew how to prove that this didn't happen, we would probably know enough to show that every digit occurs infinitely often in the decimal expansion of $\pi$, and I believe that is an open problem. So, while this scenario is almost surely false, I don't think we know enough to prove that it is false.

Now, in this scenario,
$$\cos n! = \cos (n! 2 \pi \sum_{i \geq 0} \frac{d_i}{i!}) = \cos (2 \pi N + 2 \pi \sum_{i \geq n+1} \frac{d_i }{(n+1)(n+2) \cdots i})$$
where $N$ is a very large integer. By the periodicity of $\cos$, this is just
$$\cos (2 \pi\sum_{i \geq n+1} \frac{d_i }{(n+1)(n+2) \cdots i}).$$

Now, once $n$ is large enough that $d_i$ is $0$ or $1$ for $i$ larger than $n$, we have
$$\sum_{i \geq n+1} \frac{d_i n!}{i!} \leq \frac{1}{n+1} + \frac{1}{(n+1)^2} + \frac{1}{(n+1)^3} + \cdots = \frac{1}{n}.$$
So, in our unlikely scenario, $\lim_{n \to \infty} \sum_{i \geq n+1} \frac{d_i n!}{i!} =0$ and $\lim_{n \to \infty} \cos n! = 1$.

If I may try to extract some basic lessons here, questions about trig functions evaluated at integers almost always turn into questions about $\pi$. Often they can be resolved simply by noting that $\pi$ is irrational; occasionally it is important to show that $\pi$ is not well approximated by rationals and sometimes, as in this example, they turn into claims about $\pi$ which are beyond current knowledge. The key point is to work out what they are saying about $\pi$, and then find out whether that fact is known.



Exercise: Show that, for any $a$ between $-1$ and $1$, there is a $\theta$ such that $\lim_{n \to \infty} \cos (n! \theta)$ is $a$.
I tried the following approach where I assumed the limit existed and attempted to get a contradiction. However, the best I could do was to get the same answer as David above that if (in the extremely unlikely case that) the limit exists, it will be 1. I just wanted to make a record of the approach here in case someone sees any potential to this approach.

Since $x_{n+1} = \cos((n+1)n!)$, it can be written as a $(n+1)$th degree polynomial in $\cos(n!)$. Should the limit exist, it will be a fixed point of this polynomial. Using standard multiple angle formulas (found here for ex.), we can show that the limit if it exists should equal $1$.