Homework Help: Riemann Sums - Help!

1. Apr 11, 2012

IntroAnalysis

1. The problem statement, all variables and given/known data
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].

2. Relevant equations
If P and Q1 and Q2 are partitions of [a, b], with P $\subset$ Q1, and P $\subset$ 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 $\overline{}b$) = inf{U(P,f)} so ∫(a to $\overline{}b$)≤ U(P,f)

3. 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution