Show limit of improper integral is 0

[tjkubo]

Homework Statement

Suppose $f$ is real-valued, bounded, continuous, and non-negative and suppose $\int_x^\infty f(t)\,dt$ is convergent (is finite) for all $x$. Is it true that
$$\lim_{x\rightarrow \infty} {\int_x^\infty f(t)\,dt} = 0 \ ?$$

Homework Equations

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 $\epsilon >0$, I want to show there is $M$ such that
$$\int_M^\infty f(t)\,dt < \epsilon \ .$$
I think $\lim_{t\rightarrow \infty} {f(t)} = 0$ 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.

 

[Dick]

Homework Statement

Suppose $f$ is real-valued, bounded, continuous, and non-negative and suppose $\int_x^\infty f(t)\,dt$ is convergent (is finite) for all $x$. Is it true that
$$\lim_{x\rightarrow \infty} {\int_x^\infty f(t)\,dt} = 0 \ ?$$

Homework Equations

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 $\epsilon >0$, I want to show there is $M$ such that
$$\int_M^\infty f(t)\,dt < \epsilon \ .$$
I think $\lim_{t\rightarrow \infty} {f(t)} = 0$ 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.

Try looking at the dominated convergence theorem.

 

[tjkubo]

I don't know how that would help since the set we're integrating over is changing with the limit.

 

[micromass]

Ah, so you can use that theorem??

Note that

$$\int_x^{+\infty}{f(t)dt}=\int_{-\infty}^{+\infty}{I_{[x,+\infty[}(t)f(t)dt}$$

with

$$I_{[x,+\infty[}:\mathbb{R}\rightarrow \mathbb{R}:t\rightarrow \left\{\begin{array}{cc} 0 & \text{if}~t<x\\ 1 & \text{if}~t\geq x \end{array}\right.$$

So what does the dominated convergence theorem tell you?