Q:

show that $\int \left(\sum _k |f_{k_{n}}(x)-f(x)|\right)^p \ < \infty$

let $f(n)_{n\geq1}$ be a sequence of functions in $L^P(\mathbb{R})$, $P\in[1,\infty)$ which converge to $f$ in $L^P(\mathbb{R})$.
i have choose a subsequence  $(f_{k_{n}})$ such that $$||f_{k_{n}}-f||_P \leq 2^{-2k}$$ for each $k\geq1$. Now we have to show that $$\int \left(\sum _k |f_{k_{n}}(x)-f(x)|\right)^p \ < \infty$$.
Hint is provided 
Use Holder's inequality to show $$ \left(\sum _k |f_{k_{n}}(x)-f(x)|\right)^p  \leq C \sum_k|f_{k_{n}}(x)-f(x)|^p \ 2^{kp}$$ for some fixed finite $C >0$.Use the definition of $f_{k_{n}}$ to complete the proof. 
i am confused how to use the given hint.  any help please. 
thanks in advanced

A:

\begin{align*}
\left(\int\left(\sum|f_{n_{k}}-f|\right)^{p}\right)^{1/p}&=\left(\int\lim_{N\rightarrow\infty}\left(\sum_{k=1}^{N}|f_{n_{k}}-f|\right)^{p}\right)^{1/p}\\
&=\lim_{N\rightarrow\infty}\left(\int\left(\sum_{k=1}^{N}|f_{n_{k}}-f|\right)^{p}\right)^{1/p}\\
&=\lim_{N\rightarrow\infty}\left\|\sum_{k=1}^{N}|f_{n_{k}}-f|\right\|_{L^{p}}\\
&\leq\lim_{N\rightarrow\infty}\sum_{k=1}^{N}\|f_{n_{k}}-f\|_{L^{p}}\\
&=\lim_{N\rightarrow\infty}\sum_{k=1}^{N}\dfrac{1}{2^{2k}}\\
&<\infty,
\end{align*}
where the second equality comes from Monotone Convergence Theorem and the continuity of $(\cdot)^{1/p}$.
With the hint:
\begin{align*}
\sum|f_{n_{k}}(x)-f(x)|&=\sum|f_{n_{k}}(x)-f(x)|2^{k}\cdot\dfrac{1}{2^{k}}\\
&\leq\left(\sum|f_{n_{k}}(x)-f(x)|^{p}2^{kp}\right)^{1/p}\left(\sum\dfrac{1}{2^{kq}}\right)^{1/q}\\
&=C\left(\sum|f_{n_{k}}(x)-f(x)|^{p}2^{kp}\right)^{1/p},
\end{align*}
of course, here $p>1$. For $p=1$ is trivial. 
So we have 
\begin{align*}
\int\left(\sum|f_{n_{k}}-f|\right)^{p}&\leq C\int\sum|f_{n_{k}}-f|^{p}2^{kp}\\
&=\sum\int|f_{n_{k}}-f|^{p}2^{kp}\\
&=\sum\|f_{n_{k}}-f\|_{L^{p}}^{p}2^{kp}\\
&\leq\sum 2^{-2kp}\cdot 2^{kp}\\
&=\sum 2^{-kp}\\
&<\infty. 
\end{align*}

