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.