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I am going to refer to this MSE post. In it, we are trying to show the following:

Let $\{f_n\}$ be a sequence of functions in $L^p( [0,1])$, $1 \leq p < \infty$, which converges almost everywhere to a function $f$ in $L^p$. Show that $\{f_n\}$ converges to $f$ in $L^p$ norm if and only if $\|f_n\| \to \|f\|$.

I am trying to verify the backwards implication. It is suggested that we show $\lim \sup \int |f_n-f|^{p} =0$. I tried doing this but I failed. Here is my attempt:

Since $$0 \le 2^p (|f_n|^p + |f|^p) - |f_n-f|^p \longrightarrow 2^{p+1} |f|^p,$$ Fatou's lemma tells us that $$\int 2^{p+1} |f|^p \le \lim \inf \int(2^p (|f_n|^p + |f|^p) - |f_n-f|^p) \le \lim \sup \int( 2^p (|f_n|^p + |f|^p) - |f_n-f|^p)$$ or $$2^{p+1} ||f||_p^{p} \le \lim \sup (2^p ||f_n||_p^p + 2^p ||f||^p_p - ||f_n-f||^p_p)\le 2^p \lim \sup ||f_n||_p^p + 2^p ||f_n||^p_p + \lim \sup - ||f_n-f||^p_p$$ Combining these inequalities and simplifying we get $$2^p ||f||^p_p - 2^p \lim \sup ||f_n||^p_p \le \lim \sup - ||f_n-f||^p_p$$ or $$\lim \sup ||f_n||_p^p - 2^p ||f||^p_p \ge - \lim \sup - ||f_n-f||^p_p = \lim \inf ||f_n-f||^p_p$$. Since $$||f_n||^p_p \to ||f||^p_p$$, $$\lim \sup ||f_n||^p_p = ||f||^p_p$$ and we get $$0 \ge \lim \inf ||f_n - f||^p_p...$$

But this isn't helpful (right?)...How do I prove $\lim \sup ||f_n-f||^p_p = 0$? Where

Mittens
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user193319
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    Why are you just using $\liminf \leq \limsup$ here? $$\int 2^{p+1} |f|^p \le \lim \inf \int(2^p (|f_n|^p + |f|^p) - |f_n-f|^p) \le \lim \sup \int( 2^p (|f_n|^p + |f|^p) - |f_n-f|^p)$$ That's not how you want to introduce the limsup. Start back at $$\int 2^{p+1} |f|^p \le \lim \inf \int(2^p (|f_n|^p + |f|^p) - |f_n-f|^p)$$ and try to introduce the limsup differently. – Brian Moehring Sep 17 '21 at 20:43
  • I see Marios had already given an answer, so I'll extend my previous comment: In general, we have $$\begin{align} \liminf_n a_n + \liminf_n b_n &\leq \liminf_n(a_n + b_n) \ &\leq \limsup_n a_n + \liminf_n b_n \ &\leq \limsup_n(a_n+b_n) \ &\leq \limsup_n a_n + \limsup b_n \end{align}$$ You definitely skipped one intermediate step and you essentially skipped two. That's why your method wasn't strong enough to give you the result you wanted. – Brian Moehring Sep 17 '21 at 21:03

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Note that if $a_n,b_n$ sequences in $\Bbb{R}$ and $a_n \to a$ a then $\liminf_n(a_n+b_n)=a+\liminf_nb_n$.

So

$$2^{p+1}||f||_p\leq \lim \inf_n \int2^p(|f_n|^p + |f|^p) - |f_n-f|^p=-\limsup_n\int|f_n-f|^p+2^{p+1}||f||_p$$

Which implies that $\limsup_n\int|f_n-f|^P \leq 0$

Marios Gretsas
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