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Let $[a,b]\subseteq \mathbb{R}$ be a non-degenerate closed bounded interval, and let$f,g,h:[a,b]\to\mathbb{R}$ be functions. Suppose that $f$ and $h$ are integrable, and that $\int_a^bf(x)dx=\int_a^bh(x)dx$. Prove that if $f(x)\leq g(x) \leq h(x)$ for all $x\in[a,b]$, then $g$ is integrable and $\int_a^bg(x)dx=\int_a^bf(x)dx$.

I am not allowed to use squeeze theorem. Only use the definition of Riemann integral.

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Hint: use the definition of Riemann integration and the squeeze theorem. – Quinn Culver May 14 '13 at 0:32
Simply note that the upper sum must be equal to the lower sum when it reaches the limit. – user40276 May 14 '13 at 0:33
@user40276 the integral with upper and lower sums is the Darboux integral. It's equivalent to the Riemann integral (defined using tagged partitions), but one should prove that before using it. – kahen May 14 '13 at 0:41
@kahen He's not necessarily to blame. I think I've seen more intro real analysis texts than not simply go ahead and define the Riemann integral via the Darboux definition, and some texts tragically FAIL TO INFORM the reader of this unhealthy life choice. I'm actually currently seeking a modern, rigorous purely Riemann treatment of integration and it's like hunting unicorns! – David H May 14 '13 at 1:33
@DavidH you may be interested in Neal L. Carothers's excellent book Real Analysis which covers the Darboux(–Stieltjes), Riemann(–Stieltjes) and Lebesgue (on the real line only) integrals. It even highlights (in exercises) the subtle difference between what he calls the norm integral and the Riemann–Stieltjes integral. Google Books, Amazon. – kahen May 14 '13 at 1:47

Write $U(P, g)$ and $L(P, g)$ to denote the lower and upper sums for function $g$ over specified partition $P$ of $[a,b]$. Pick a partition $P_1$ and $P_2$ such that $$ U(P_1, h)-\int_a^b h(x)dx<\epsilon$$ and $$ \int_a^b f(x)dx-L(P_2, f)<\epsilon$$ This is possible because both $f$ and $h$ are Riemann integrable. Let $P = P_1\cup P_2$ (common refinement of $P_1$ and $P_2$). Observe that $f(x)\le g(x)\le h(x)$ imply $$ U(P, g)-L(P, g)\le U(P, h) - L(P, f)\le U(P_1, h)-L(P_2, f) < 2\epsilon $$ This shows that $g$ is Riemann integrable. Can you proceed from here?

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Can you just use the definition of Riemann integral instead of upper sums and lower sums? – SAS May 14 '13 at 1:20
The definition of Riemann integral I learned (from Rudin's "Principles of Mathematical Analysis") involves upper and lower sums. What other definition do you have in mind? – Prism May 14 '13 at 1:21

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