Proving Integrability of Bounded Functions using Riemann Sums - Homework Help

[IntroAnalysis]

Homework Statement

Suppose f:[a,b] → ℜ is bounded and for each ε > 0, ∃ a partition P such that for any refinements Q1 and Q2 of P, regardless of how marked ⎟S(Q1,f) – S(Q2,f)⎟ < ε. Prove that f is integrable on [a,b].

Homework Equations

If P and Q1 and Q2 are partitions of [a, b], with P [itex]\subset[/itex] Q1, and P [itex]\subset[/itex] Q2, then Q1 and Q2 are refinements of P

∫(a to b) = sup {L(P,f)} so L(P,f) ≤ ∫(a to b)

∫(a to [itex]\overline{}b[/itex]) = inf{U(P,f)} so ∫(a to [itex]\overline{}b[/itex])≤ U(P,f)

The Attempt at a Solution

I know the S(Q1,f) and S(Q2,f) are squeezed between the Lower sum (P,f) and the Upper sum (P,f), and that the upper and lower Riemann integrals are less than or equal to the Upper and Lower sums. To prove f is integrable on [a,b] I have to show f is bounded, that was given, and show the upper and lower Rieman integrals are equal.

 

[lippyka]

I am not sure how to show that the Upper and Lower Riemann Integrals are equal. I know if I can show U(P,f) - L(P,f) < ε for any ε > 0, then the integrals would be equal. But how do I use the information given that S(Q1,f) – S(Q2,f) < ε, regardless of how marked?