Let $R$ be a rectangle of $\mathbb{R}^n$ and let $f:R \to \mathbb{R}$ bounded. Assume that there exists a sequence $\{P_n\}$ of partitions of $R$ such that $$\lim_n [S(f,P_n)-s(f,P_n)]=0$$ Show that $f$ is integrable in $R$, that ${s(f,P_n)}$ is convergent and that $$\int_R f = \lim_n s(f,P_n)$$
I think I'm struggling with the definitions of upper and lower sums. For the part when the questions says to prove integrability, since the limit of ${S(f,P_n)-s(f,P_n)}$ exists and equals $0$ then for each $\varepsilon >0$, there is some $N\in \mathbb{N}$ such that if $n\geq N$, then $$S(f,P_n)-s(f,P_n)<\varepsilon$$ Now taking $P_N$, the inequality condition is the same as the Riemann integrability criterium, so $f$ is integrable in $R$.
For the last two parts, I have no clue about how should I proceed (I'm not even sure that what I wrote above holds), so I need some help. I've not used yet the hypothesis that $f$ is bounded, but I can't see what the rol of being bounded is. Thanks in advance!
The boundedness of $f$ allows the definition of lower and upper sums.
After proving the first part, note that $$s(f,P_n) \le \int_R f \le S(f,P_n)$$ for every $n$, so $$s(f,P_n)-S(f,P_n) \le \int_R f - S(f,P_n) \le \int_R f - s(f,P_n) \le S(f,P_n)-s(f,P_n)$$ and then $$\left|\int_R f - s(f,P_n)\right| \le S(f,P_n)-s(f,P_n)$$