a function $f$ is analytic on an open set $ \Omega$ if for every compact $K \subset \subset \Omega $ there exist a constant $c>$ such that :
$$\sup_K|f^{(n)}(x)| \leq c^{n+1}n!, \quad \forall n \in \mathbb{N}$$
can someone recommend a reference for the proof please
This is immediate from Taylor's Theorem. One wonders why you want that "reference". If, say, you're writing a research paper you don't need a reference, simply saying "Taylor's theorem shows that..." should be more than enough justification.
If otoh you want a reference with a solution to your homework you should work it out for yourself. Hint: Consider the Lagrange form of the remainder; if $|x-a|<1/c$ then $c^n(x-a)^n\to0$ as $n\to\infty$.