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I want to show that $\underline{\int_{a}^{b}} f \ d \alpha \leq \overline{\int_{a}^{b}} f \ d \alpha$

So I want to show that $\sup L(P,f, \alpha) \leq \inf \ U(P, f, \alpha)$. Can I just suppose that $\sup L(P,f, \alpha)> \inf \ U(P, f, \alpha)$ and come up with a contradiction?

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    You could try to prove directly that L(P,f,alpha) is at most U(Q,f,alpha) for every given P and Q, then conclude. No proof by contradiction is needed here.2011-08-26
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    @Didier Piau: But proof by contradiction seems to be easier.2011-08-26
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    @Ryan: It is unclear how you determine relative ease without having an argument by contradiction.2011-08-26
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    Relevant: http://math.stackexchange.com/questions/59606/upper-and-lower-sums-in-riemann-integral2011-08-26
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    @Dylan Moreland: What made you change your mind about Michigan?2011-08-26
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    @Ryan I got my undergraduate degree there and moved on; Michigan is a great place! This is very far off-topic here, so find me in chat if you want to know more.2011-08-26
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    @Ryan, maybe it seems to be (to you) but it is not. In fact it is a very healthy practice, once you finished concocting a proof, to check whether it is possible to rewrite the parts (if any) where you used contradiction, without this twist. Often, it is quite possible to do so and the resulting proof is much simpler than the original one. The case at hand is a good example of this phenomenon.2011-08-27

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