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## Homework Statement

Hi All, I've been having great difficulty making progress on this problem.

Suppose g

_{n}converges to g a.e. on [0,1]. And, for all n, g

_{n}and h are integrable over [0,1]. And |g

_{n}|[tex]\leq[/tex]h for all n.

Define G

_{n}(x)=[tex]\int[/tex]gn(x) from 0 to x.

Define G(x)=[tex]\int[/tex]g(x) from 0 to x.

**Prove:**Gn converges to G

**uniformly**.

**2. The attempt at a solution**

So, here's what I've got. We can use dominated convergence theorems to show that Gn goes to G pointwise on [0,1]. Since the set is compact, this should also provide some insights.

Moreover, I was thinking we could use Egorov's theorem to take away the set on which G does not go to G uniformly. I'm not sure how to get rid of that otherwise.