TITLE: A uniformly convergent sequence of real analytic functions which does not converge to a real analytic function
QUESTION [5 upvotes]: 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!

REPLY [2 votes]: 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 $(1+x^2)^{-1}$.