Here is the Brezis-Lieb Lemma:
$(X,\mathfrak{A},\mu)$ is a measure space, consider $L^{p}(X):p \in(0,\infty)$ . Then $\left\{ f_{n} \right\}$ is a sequence of extended complex-values measurable functions on $L^{p}(X)$ such that
1）$\left\{ f_{n} \right\}$ is uniformly bounded by $p$ norm: i.e. $\exists M \geq 0, \forall n \in \mathbb{N} :\|f_{n}\|_{p} \leq M<\infty$
2） $f_{n}\to f,\mathrm{\; a.e. \;} X$
Then we have
1）$f \in L^{p}$
2） $$\begin{align*} \lim\limits_{n\to \infty } \int _{X} |f_{n}|^{p} - |f|^{p} - |f_{n}-f|^{p} \ d\mu =0 \end{align*}$$
I got stuck in one step of proving the Lemma
In one step, it showed that given any $\epsilon$, there exists a positive constant $C_{\epsilon,p}$ such that $$\begin{align*}
\left| |f_{n}|^{p}-|f|^{p} -|f_{n}-f|^{p} \right| &\leq \left| |f_{n}|^{p} - |f_{n}-f|^{p} \right| +|f|^{p}
\\ &\leq \epsilon|f_{n}-f|^{p}+C_{\epsilon,p} |f|^{p} +|f|^{p}
\\ &= \epsilon|f_{n}-f|^{p}+( C_{\epsilon,p}+1) |f|^{p}
\\ \left| |f_{n}|^{p}-|f|^{p} -|f_{n}-f|^{p} \right|- \epsilon|f_{n}-f|^{p} &\leq (C_{\epsilon,p}+1)|f|^{p}
\end{align*}$$
Then it's clear that $(C_{\epsilon,p}+1)|f|^{p}$ is a $\mu$ integrable function on $X$ when $C_{\epsilon,p}$ is fixed . Then I hope I can use the Lebesgue Dominant Convergence Theorem. However, the theorem requires that $$\begin{align*}
\left| \ \left| \ |f_{n}|^{p}-|f|^{p} -|f_{n}-f|^{p} \right|- \epsilon|f_{n}-f|^{p} \right| \leq (C_{\epsilon,p}+1)|f|^{p}
\end{align*}$$ How can I show this?
I guess one should use the fact $\epsilon$ can be arbitrarily small, so we have $$\begin{align*}
\left| |f_{n}|^{p}-|f|^{p} -|f_{n}-f|^{p} \right|- \epsilon|f_{n}-f|^{p} \geq 0
\end{align*}$$ but I can't show this.
Any help with this? Thanks!
