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Let $f$ and $g$ be continuous functions on $(a,b)$ such that $0 \le f\left( x \right) \le g(x)$ for all $x \in \left( {a,b} \right)$; $a$ can be $ - \infty $ and $b$ can be $ + \infty $.

Prove: $\mathop \int \limits_a^b g\left( x \right)dx < \infty \Rightarrow \mathop \int \limits_a^b f\left( x \right)dx < \infty $

This is an exercise in Elementary Analysis by Ross. Chapter 36, exercise 6a. I think I'm almost able to prove this, using the definitions give in 36 and some theorems in 33. The thing is that I don't use that $0 \le f\left( x \right)$. Could anybody help me where in the proof I need to use this?

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    Wow.. brilliant. Thank you so much :) It's sad that I can't upvote you, I really appreciate your answers.2012-12-11

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