Update: This question differs from Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition? in the following way:
From what I see in that question(and answer by Thorgott) they assume that they have a sequence of partitions, and show that if the Riemann sum converges for all selctions it must be the same for another selection(with the same sequence of partitions). They do not show that if we have a given sequence of partitions that converges for all selections(for that sequence of parititons), than any other sequence of partitions with any selection also converges to the same number.
Start of original post:
I have this definition of a Riemann-sum:
Given a $f: [a,b]\rightarrow \mathbb{R}$ and a partition$\Pi=\{a=x_0,x_1,\ldots,x_n=b\}$ and a selction $C=\{c_1,c_2,\ldots,c_n\}$ where$c_i\in [x_{i-1},x_i]$. We define the Riemann-sum
$$R(\Pi,C)=\sum\limits_{i=1}^n f(c_i)(x_i-x_{i-1}).$$Define $\lvert\Pi\rvert = \max \{x_i-x_{i-1}:1\le i\le n \}$
We define the Riemann-integral:
The function $f:[a,b]\rightarrow \mathbb{R}$ is Riemann-integrable ifthere is a real number $\alpha$ such that
$$\lim\limits_{n \rightarrow \infty} R(\Pi_n,C_n)=\alpha,$$
for every sequence $\{\Pi_n,C_n\}$ of partitions and selctionssuch that $\lvert\Pi_n\rvert \rightarrow 0$. The value $\alpha$ isthen valled the Riemann-integral $\int_a^b f(x) dx$.
My question is: Is the definition below sufficient?:
The function $f:[a,b]\rightarrow \mathbb{R}$ is Riemann-integrable ifthere exists a sequence of partitions $\{\Pi_n\}_{n=1}^\infty,$ and a real number $\alpha$, such that $\lvert \Pi_n \rvert \rightarrow 0$ and such that for any selection $C_n$ we have
$$\lim\limits_{n \rightarrow \infty} R(\Pi_n,C_n)=\alpha,$$
for every sequence $\{C_n\}$. The value $\alpha$ isthen valled the Riemann-integral $\int_a^b f(x) dx$.
Basically: If we get it to work for one sequence of partitions, does it then converge for all sequence of partitions where the norm goes to zero?