$\mathbf {P}$ is a probability measure on $(\mathbf {R},\mathcal{B}).$ Show that for any $p\in[1,\infty]$, the class of all bounded continuous functions is a dense subset of $L^p (\mathbf {R},\mathcal{B},\mathbf {P}) $
I have proved it for $p\in[1,\infty)$, in the process I have shown that indicator function of any measurable set can be arbitrarily approximated by bounded continuous functions. But I am struggling in the $\infty $ case. Any help please.
Here $||f||_\infty=$ inf $\{\lambda : P (|f|>\lambda)=0\}$