Can Convergent Upper and Lower Riemann Sums Establish Integrability?

[jdz86]

Homework Statement

Suppose f:[a,b] \rightarrow \Re is bounded and that the sequences {U_{P_{n}}(f)}, {L_{P_{n}}(f)} are covergent and have the same limit L. Prove that f is integrable on [a,b].

Homework Equations

U_{P_{n}}(f) is the upper sum of f relative to P, and L_{P_{n}}(f) is the lower sum of f relative to P, where in both cases P is the partition of [a,b].

Also a hint was to examine the sequence {a_{n}} where a_{n} = U_{P_{n}}(f) - L_{P_{n}}(f)

The Attempt at a Solution

Didn't figure out the full proof, only parts:

I started with having P = {x_{0},..., x_{n}}, where x_{0} = a, and x_{n} = b. And given that they are both convergent and have the same limit, using the hint I can show that they both converge to a_{n}, but I got stuck after that.

 

[JG89]

It's integrable if a single value exists between the two sequences as n tends to infinity, right? I'd look at this as a nested sequence of intervals, where the length of the nth interval as n approaches infinity is 0 (because the two sequences have the same limit), and from there it should be easy.