Closed subset of $L^1([0,1])$

Closed subset of $L^1([0,1])$

I want to show this:
Consider $L^{1}([0,1])$ and $L^{2}([0,1])$, and $$G=\left\{ f
\;\middle\vert\; \int_{0}^{1} |f|^2 \,dm \leq n \right\}\subseteq
L^{2}([0,1]),$$ for some $n\in\mathbb{N}$. Show that $G$ is closed in
$L^{1}([0,1])$ with the norm of $L^{1}([0,1])$. The measure being used
here is Lebesgue measure.
I tried Minkowski and Hölder, but got nothing. Thanks for your help.