Separability of Lp spaces

I should show the following statements:

1. $L^\infty(\mathbb{R}^n, \delta_{x_0})$ is separable;
2. $L^p(\mathbb{R}^n, \#)$ is not separable for any p, where $\#$ is the counting measure on $\mathbb{R}^n$. Can someone help me, please?