(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose [itex]f[/itex] is real-valued, bounded, continuous, and non-negative and suppose [itex]\int_x^\infty f(t)\,dt[/itex] is convergent (is finite) for all [itex]x[/itex]. Is it true that

[tex]\lim_{x\rightarrow \infty} {\int_x^\infty f(t)\,dt} = 0 \ ?[/tex]

2. Relevant equations

3. The attempt at a solution

I can't think of a counterexample and it seems true intuitively, so I'm trying to prove it's true.

Given [itex]\epsilon >0[/itex], I want to show there is [itex]M[/itex] such that

[tex]\int_M^\infty f(t)\,dt < \epsilon \ .[/tex]

I think [itex]\lim_{t\rightarrow \infty} {f(t)} = 0[/itex] although I am not sure how to prove this. Specifically, how would you prove that the improper integral would not exist if this limit did not exist?

I'm not sure what to do at this point.

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# Show limit of improper integral is 0

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