Solve Riemann Integral: Show \( \int_a^b f = \lim U_n = \lim L_n \)

[Fantini]

Greetings everyone. First, it's great that the site is back again and I hope it can be merged soon enough. :D

Here's the question:
Let \( f \) be a bounded function on \( [a,b] \). Suppose there exist sequences \( (U_n) \) and \( (L_n) \) of upper and lower Darboux sums such that \( \lim (U_n - L_n) = 0 \). Show that \( f \) is integrable and that \( \int_a^b f = \lim U_n = \lim L_n \).

Here's my try:

By the hypothesis, exists \( M > 0 \) such that for all \(n > M, \varepsilon > 0 \) we have \( | U_n(f,P) - L_n(f,P) | < \varepsilon \), hence \( U_n(f,P) - L_n(f,P) < \varepsilon \) for some partition \( P \) of \( [a,b] \). It follows then that \( f \) is integrable, and by the limit properties we see that \( \lim(U_n - L_n) = \lim U_n - \lim L_n = 0 \implies \lim U_n = \lim L_n \).

My question is if that wouldn't imply already that \( \lim U_n = \int_a^b f \)? If not, I'm a bit lost. Would I have to show that for all \( n > M \) we have that \( L_n [f] \geq U_n [f] \), where \( L_n [f] \) and \( U_n[f] \) mean the lower and upper Darboux integrals respectively?

Also, that awkward moment when you type ( f ) without spaces and it becomes (f). (Tongueout)

 

[Plato2]

Here's the question:
Let \( f \) be a bounded function on \( [a,b] \). Suppose there exist sequences \( (U_n) \) and \( (L_n) \) of upper and lower Darboux sums such that \( \lim (U_n - L_n) = 0 \). Show that \( f \) is integrable and that \( \int_a^b f = \lim U_n = \lim L_n \).

This wording seems odd to me. Darbuox sums involve a partition of $[a,b]$.
So when the question says that $U_n~\&~L_n$ are upper and lower sums are we to assume that there is a partition $P_n$ associated with each pair? Moreover, is seems that $P_{n+1}$ should be a refinement of $P_n$
Is that mentioned in the statement of the question?

 

[Fantini]

No, I copied the problem as it's written. It's from the book "Elementary Analysis: The Theory of Calculus" by Kenneth Ross. I picked it up at the library as an option to my current analysis course and enjoyed it so far.

I admit that the thought that each \( U_n \& L_n \) would have a partition \( P_n \) associated with the pair occurred to me, but I decided not to follow that path.