Convergence in Lp from ||fn|| to ||f|| with almost everywhere convergence

[Jamin2112]

Homework Statement

As part of a larger proof, I'm trying to show that ||f_n|| --> ||f|| implies <f_n> --> f in L_p, where <f_n> is a sequence of functions in L_p, 1≤p<∞, which converge a.e. to a function f in L_p.

Homework Equations

||f|| = (∫|f|^p)^1/p.

The Attempt at a Solution

There's some string of inequalities I need to obtain ||f-fn||≤|||f||-||fn|||<∂. Any ideas?

 

[micromass]

Apply Fatou's lemma on

$$\lim_{n\rightarrow +\infty} {2^p(|f_n|^p+|f|^p)-|f_n-f|^p}$$

 

[Jamin2112]

Apply Fatou's lemma on

$$\lim_{n\rightarrow +\infty} {2^p(|f_n|^p+|f|^p)-|f_n-f|^p}$$

Genius!