How to mathematically prove that a relation T(n)=T($\sqrt{n}$)+c is O(log(log(n))?
Let $x = \log n$ and $Q(x) = T(2^x)$. You can rewrite your recurrence as follows:
$$
Q(x) = T(2^x) = T(n) = T(n^\frac{1}{2}) + c = T(2^{x/2}) + c = Q(x/2) + c.
$$

Which is easily solved using, e.g., the Master Theorem to obtain $Q(x) = \Theta(\log x)$. Substituting back:
$$
T(n) = Q(x) = \Theta(\log x) = \Theta(\log \log n).
$$