TITLE: Is it true that for each bounded continuous function we can find a set of analytic functions to uniformly converge it?
QUESTION [8 upvotes]: Is it true that for each bounded continuous function $f:\mathbb R \to \mathbb R$, we can find a set of analytic functions $g_i:\mathbb R \to \mathbb R, i=1,2,...$ such that $g_i$ uniformly converges to $f$ ?

REPLY [8 votes]: In the paper 

MR0098847 (20 #5299)
Grauert, Hans:
On Levi's problem and the imbedding of real-analytic manifolds. 
Ann. of Math. (2) 68 1958 460–472. 

it is proved (Proposition 8) that real analytic functions are dense in continuous functions for the Whitney $C^0$-topology, for any paracompact real analytic manifold. The sup-norm gives a coarser topology, so this also follows.