# Need help with part of a proof (measure theory class)

1. May 29, 2012

### Jamin2112

1. The problem statement, all variables and given/known data

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

2. Relevant equations

||f|| = (∫|f|p)1/p.

3. The attempt at a solution

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

2. May 29, 2012

### micromass

Staff Emeritus
Apply Fatou's lemma on

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

3. May 29, 2012

Genius!