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## Main Question or Discussion Point

I ran into this proof in one of my textbooks and was wondering if anybody could lead me in the right logical direction. I can prove the first-differentiable continuous case but the infinity case throws me off. Please help if you can!

Thanks!

Suppose [tex]\left\{f}\right\}\subset C_{\infty}\left(\left[a,b\right]\right)[/tex] such that [tex]\left{f\right}_{n}[/tex] converges uniformly to some [tex]\left{f\right}\in C_{\infty}\left(\left[a,b\right]\right)[/tex]. Prove that:

[tex]\int^a_b\left{f\right}_{n}\left(x\right)dx \rightarrow \int^a_b\left{f\right}\left(x\right)dx[/tex]

Thanks!

Suppose [tex]\left\{f}\right\}\subset C_{\infty}\left(\left[a,b\right]\right)[/tex] such that [tex]\left{f\right}_{n}[/tex] converges uniformly to some [tex]\left{f\right}\in C_{\infty}\left(\left[a,b\right]\right)[/tex]. Prove that:

[tex]\int^a_b\left{f\right}_{n}\left(x\right)dx \rightarrow \int^a_b\left{f\right}\left(x\right)dx[/tex]