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 $f$ ?
Convolve it with narrower and narrower Gauss kernels.
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.