# Proving weak convergence

Suppose $$\text{sup}\|f_n\|_p <+ \infty$$ and $$f_n \to f$$ almost everywhere on $$E$$. I want to prove weak convergence $$f_n \rightharpoonup f.$$

I already have a proof that there exists a sub-sequence $$\{f_{n_i}\}$$ such that $$f_{n_i}\rightharpoonup f$$ weakly in $$L^p.$$

So in can reformulate my question as:

How to show from the weak convergence of such sub-sequence the convergence of the whole sequence?

You have to show that $$\int f_ng \to \int fg$$ for all $$g \in L^{q}$$ where $$\frac 1 p+\frac 1 q=1$$. For this it is enough to show that every subsequence of $$(\int f_ng )$$ has a subsequence converging to $$\int fg$$.

A sequence of real numbers $$(a_n)$$ converges to a real number $$a$$ iff every subsequence of $$(a_n)$$ has a further subsequence converging to $$a$$. [You can give a proof by contradiction].

• can you expand a little? I don't understand.
– user733151
Dec 9 '19 at 12:25
• @ann.boz I added some details when you were typing the comment. Dec 9 '19 at 12:26
• and how does this prove what I want?
– user733151
Dec 9 '19 at 12:26
• I did not prove that every subsequence weakly converges, I just proved that there exists one that does so. So I don't see how you proved what I want to prove.
– user733151
Dec 9 '19 at 12:29
• I'm sorry I'm not understanding.. :(
– user733151
Dec 9 '19 at 12:30