A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function

Question

I am looking for an example of a uniformly convergent sequence of real analytic functions which does not converge to a real analytic function. Also I would appreciate any pointers on how to think of (picture) real analytic functions. Thanks!

Answer 1

Choose your favorite continuous, non-real-analytic function on a compact interval, and then apply Weierstrass' approximation theorem to it. Bernstein polynomials can give a constructive example if you want.

Answer 2

If you are content with something on a bounded interval take $(x \tanh(n x))_n$ this will converge uniformly to $|x|$ on every bounded interval. If you want it uniform on all of $\mathbb{R}$ multiply by $(1+x^2)^{-1}$.