$L^{p}$ convergence equivalent condition I have to show that for $p\in[0,\infty)$, $f_{n},f\in L^{p}(\mathbb{R})$, (i) $f_n\rightarrow f$ in $L^{p}([-N,N])$ for all $N\in \mathbb{N}$ and (ii) $\lvert\lvert f_{n}\rvert\rvert \rightarrow \lvert\lvert f\rvert\rvert$ implies $f_{n}\rightarrow f$ in $L^{p}(\mathbb{R})$.
My approach was to first deconstruct $\lvert\lvert f_{n}-f\rvert\rvert_{p}^{p}$ as
$$
\lvert\lvert f_{n}-f\rvert\rvert_{p}^{p}=\int_{-\infty}^{-N}\lvert f_{n}-f\rvert^{p}+\int_{-N}^{N}\lvert f_{n}-f\rvert^{p}+\int_{N}^{\infty}\lvert f_{n}-f\rvert^{p}.
$$
(The integral is taken w.r.t. the Lebesgue measure). By (i) we know that the above decomposition is valid for all $N\in \mathbb{N}$ and as we take the limit $n\rightarrow \infty$, the middle term vanishes. The problem is how to get an upper bound $\epsilon_{n}$ for the left and right integral s.t. $\epsilon_{n}\rightarrow 0$ as $n\rightarrow \infty$. I don't see how we could use (ii) for that. I tried to show that  for all $n$ sufficiently large we get an $N^{*}$ such that the left and right integral are bounded and that this bound approaches $0$ as $n \rightarrow \infty$, but didn't succeed. Could anyone help?

We can show it using the following facts.

*

*If $\left(g_n\right)_{n\geqslant 1}$ is a sequence such that $g_n\to g$ almost everywhere and $\int \lvert g_n\rvert^p\to\int \lvert g\rvert^p$, then $\int \lvert g_n-g\rvert^p\to 0$.


*If $(f_n)_{n\geqslant 1}$ satisfies the conditions of the opening post, then there exists a subsequence $(g_n)=(f_{\varphi(n))})$ where $\varphi\colon\mathbb N\to\mathbb N$ is increasing, such that $\int \lvert g_n-f\rvert^p\to 0$.
In order to show fact one, we can use Fatou's lemma to the sequence $(h_n)$ given by
$$
h_n=
\begin{cases}
\lvert f\rvert^p+\lvert f_n\rvert^p-\lvert f_n-f\rvert^p&\mbox{ if }0<p\leqslant 1,\\
2^{p-1}\lvert f\rvert^p+2^{p-1}\lvert f_n\rvert^p-\lvert f_n-f\rvert^p&\mbox{ if } p\geqslant 1.
\end{cases}
$$
For fact 2., we use the fact that convergence in $L^p$ implies the almost sure convergence of a subsequence combined with a diagonal extraction argument.
To conclude, fact 2. applied to a subsequence of $(f_n)$, say $(f_{\psi(n))})$  show that there exists a further subsequence $(f_{\varphi\circ \psi(n)})$ such that $\int\left\lvert f_{\varphi\circ \psi(n)}-f\right\rvert^p\to 0$.
