Riemann Integral

Fantini

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

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).

Plato

Well-known member
MHB Math Helper
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

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.