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## Homework Statement

[tex]\int_{a}^\infty\ f(x) dx [/tex] <--- converge

f(x) uniformly continuous in [a,[tex]\infty[/tex]]

prove that [tex]lim_{x\rightarrow \infty} f(x) = 0[/tex]

## Homework Equations

## The Attempt at a Solution

I know that if f(X) has a limit in [tex]\infty[/tex] it has to be 0

I think that the solution has to be conected to the fact that if f(x) uniformly continuous ,there is a M that |f'(x)|<M,

I think I can prove that if f(x) does not have limit it's Derivative has to change infinite times form + to - , so f(x) has to go up and down infinite times..

and when it go up there is a limit to how low her max can be,

I think I have to put it all together with Cauchy test

But I can't seem to do it.

Thank you