I'm learning about measure theory (specifically Lebesgue integration) and need help with the following problem:
Let $f_n, f \in L^1$ and $\int_{\mathbb{R}}\left|f_n-f\right| \rightarrow0$. Prove that $(1)$ $\int_{\mathbb{R}}f = \lim_{n\to\infty} \int_{\mathbb{R}}f_n$ and $(2)$ $\int_{\mathbb{R}}\left|f\right| = \lim \int_{\mathbb{R}}\left|f_n\right|$
I have found a solution for $(2)$ using the reverse triangle inequality but I don't understand how it works.
$(2)$ If $\int_{\mathbb{R}}\left|f_n-f\right| \rightarrow0$ then by the reverse triangle inequality $$0 \leq \left| \int_{\mathbb{R}}\left|f\right| - \int_{\mathbb{R}}\left|f_n\right|\right| \leq \int_{\mathbb{R}}\left|f_n-f\right|.$$ Therefore $\lim_{n\to\infty} \int_{\mathbb{R}}\left|f_n\right| = \int_{\mathbb{R}}\left|f\right|$.
I don't understand how the reverse triangle inequality works here using the assumption that $\int_{\mathbb{R}}\left|f_n-f\right| \rightarrow0$. Could someone explain this in more details? How do I show $(1)$?
For $(1)$ note that $$ \left| \int_{\mathbb{R}}f - \int_{\mathbb{R}}f_n\right| \leq \int_{\mathbb{R}}\left|f_n-f\right|. $$ For $(2)$ note that by this inequality $$ \left| \int_{\mathbb{R}}\left|f\right| - \int_{\mathbb{R}}\left|f_n\right|\right| \leq \int_{\mathbb{R}}\Big||f_n|-|f|\Big|\leq \int_{\mathbb{R}}\left|f_n-f\right|. $$ Letting $n\to\infty$ we obtain both properties, since $$ \lim_{n\to\infty}\left| \int_{\mathbb{R}}f - \int_{\mathbb{R}}f_n\right|=\left| \int_{\mathbb{R}}f - \lim_{n\to\infty}\int_{\mathbb{R}}f_n\right| $$ and $$ \lim_{n\to\infty}\left| \int_{\mathbb{R}}\left|f\right| - \int_{\mathbb{R}}\left|f_n\right|\right| =\left| \int_{\mathbb{R}}\left|f\right| - \lim_{n\to\infty}\int_{\mathbb{R}}\left|f_n\right|\right|. $$