Can someone solve my non-understandable process in proving a theorem? Theorem.
Let $E$ be a subset of $\mathbb{R}^n$.
Then, if $p\gt0$, $\int_E|f-f_k|^p\to0$, and $\displaystyle\int_E|f_k|^p\le{}M$ for all $k$, then $\displaystyle\int_E|f|^p\le{}M$.
For your information, $|\cdot|$ means a Lebesgue measure.



*

*There is a Theorem A, $$\int_E|f|^p=p\int_0^\infty\alpha^{p-1}w_{|f|}(\alpha)d\alpha$$
,where $w(\alpha)=\left|\left(\mathbf{x}\in{}E:f(\mathbf{x})\gt\alpha\right\}\right|$

*Since $|f|\le|f-f_k|+|f_k|$ by the triangle inequality,
$$\int_E|f|\le\int_E\left(|f-f_k|+|f_k|\right)=\int_E|f-f_k|+\int_E|f_k|$$

*From the Theorem A when $p=1$,
$$\int_0^\infty w_{|f|}(\alpha)d\alpha \le \int_0^\infty w_{|f-f_k|}(\alpha)d\alpha + \int_0^\infty w_{|f_k|}(\alpha)d\alpha$$

*Since (?),
$$w_{|f|}(\alpha) \le w_{|f-f_k|}(\alpha) + w_{|f_k|}(\alpha)$$

*Thus,
\begin{align}
\int_E|f|^p &= p\int_0^\infty\alpha^{p-1}w_{|f|}(\alpha)d\alpha \\
&\le p\int_0^\infty\alpha^{p-1}\left(w_{|f-f_k|}(\alpha) + w_{|f_k|}(\alpha)\right)d\alpha \\
&= p\int_0^\infty\alpha^{p-1} w_{|f-f_k|}(\alpha) d\alpha + p\int_0^\infty\alpha^{p-1} w_{|f_k|}(\alpha) d\alpha \\
&=\int_E|f-f_k|^p + \int_E|f_k|^p
\end{align}

*Letting $k\to\infty$,
$$\int_E|f|^p \le M$$



I do not understand how the third process causes the fourth process.
Also, I do not understand the last process.
We are proving that $\displaystyle\int_E|f|^p\le{}M$, not proving that $\displaystyle\int_E|f|^p\le{}M$ as $k\to\infty$.

*

*Since $p >0$, we have that
$$ |f-g|^p \leq |f|^p + |g|^p $$

*Using 1, we see
$$ \int |f|^p dx \leq \int | f- f_k|^p dx + \int |f_k|^p dx $$
for all $k \in \mathbb{N}$.

*Apply bound, we see
$$\int |f|^p dx \leq \int | f- f_k|^p dx + M $$

*Apply the limit, we see
$$\int |f|^p dx \leq\underbrace{ \lim_{k \to \infty} \int | f- f_k|^p dx}_{=0}+ M \leq M $$
as desired.

