Question about $L^{1}$ and $L^\infty$ spaces

Question

I need help showing that $L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})$ is dense in $L^1$ but it isn't dense in $L^\infty(\mathbb{R})$. I really don't have a clue how to do this so any help is appreciated.

Answer 1

Take any integrable $f$. By Lebesgue's dominated convergence we have $$\lim_{n\to\infty}\int |f-f\cdot 1_{|f|\leq n}| = 0 $$ but $f\cdot 1_{|f|\leq n}$ belongs to $L^1\cap L^\infty$, so it's dense in $L^1$.

For the second one, take $f\equiv 1$. Then suppose there is $g\in L^1\cap L^\infty$ so that $$||f-g||_\infty<1/2 $$ then $$1/2<g<3/2\ a. e. $$but this is contradiction with the fact that $g\in L^1$. So $||f-g||_\infty\geq 1/2$ for any such $g$, so $f$ can't be approximated by elements of $L^1\cap L^\infty$, which means it's not dense in $L^\infty$.