Theorem: Suppose $ f \in R([a,b]) $ with integral $ I = \int_{a}^{b} f $. Then for any sequence of partitions $ \{ \prod_n \}_{n=1} $ s.t. $ \lim \lambda(\prod_n) = 0 $ and for every sequence of finite sequence of appropriate numbers $\left\{ \{t_{i} \}_{i=1}^{(n)}\right\}_{n=1}^{M_{n}}$ ( where $ \forall n ,i. t_i^{(n)} \in \prod_n $ and $ M_n $ is the number of points in the $n $ partition), then:
$S\left(f, \Pi_{n},\left\{t_{i}^{(n)}\right\}_{i=1}^{M_{n}}\right) \rightarrow I$
Notes about notation:
- $ f \in R([a,b]) $ means $ f$ is Riemann integrable on $[a,b] $ .
- ( mesh of partition ) $ \lambda(\prod _n) = max_{i=1,…,M_n}|{ \triangle x_i}| $ , where $ x_i \in \prod_n $, for all $ 1 \leq i \leq M_n $, where $ M_n $ is the number of points in the partition.
- ( Riemann sum ) given $ f:[a,b] \rightarrow \mathbb{R} $ a partition $ \prod $ and finite sequence of approproiate numbers $ \{ t_i \} $, their Riemann sum is defined as: $S\left(f, \Pi,\left\{t_{i}\right\}\right)=\sum_{i} f\left(t_{i}\right) \Delta x_{i}$
- ( finite sequence of appropriate numbers ) Given a partition $ \prod = ( x_0,…,x_l ) $, we'll say $ \{ t_1,…,t_l \} $ are appropriate points for the partition if it occurs that $t_i \in [ x_{i-1} , x_i] $ for all $ i = 1,…,n $
Attempt :
Let $ \{ \prod_n \}_{n=1} $ be arbitrary sequence of partitions of $ [a,b] $ s.t. $ \lim \lambda(\prod_n) = 0 $. Let $ \left\{ \{t_{i} \}_{i=1}^{(n)}\right\}_{n=1}^{M_{n}} $ be an arbitrary sequence of finite sequence of appropriate numbers s.t. $ \forall n \geq 1. 1 \leq i \leq M_n. t_{i}^{(n)} \in \prod_n $ and $ M_n $ is the number of elements in partition $ \prod_n $. Since $ f $ is Riemann integrable on $ [ a,b] $ then we know that for every $ \epsilon > 0 $ there exists $ \delta > 0 $ s.t. for every partition $ \prod $ with $ \lambda(\prod) < \delta $ and for every appropriate finite sequence of elements $ \{ t_i \} $ of the partition then $ | S(f,\prod , \{ t_i \} ) – I | < \epsilon $ .
We'll show that $ \forall \epsilon>0. \exists N \in \mathbb{N}. \forall n> N . | S\left(f, \Pi_{n},\left\{t_{i}^{(n)}\right\}_{i=1}^{M_{n}}\right) – I | < \epsilon $. Let $ \epsilon >0 $ be arbitrary, [ rest of proof goes here ] .
From assumption , I know I have partitions $ \prod_1,…,\prod_n $ s.t. $ \lim \lambda(\prod_n) = 0 $, I have to use the fact that for every $ \epsilon > 0 $ there exists $ \delta > 0 $ s.t. for every partition $ \prod $ with $ \lambda(\prod) < \delta $ and for every appropriate finite sequence of elements $ \{ t_i \} $ of the partition then $ | S(f,\prod , \{ t_i \} ) – I | < \epsilon $ . The problem is, I don't know how to use $ \lim \lambda(\prod_n) = 0 $ in order to take advantage of the fact that $ f $ is Riemann integrable. Do you have any ideas as how to prove the theorem? I don't know what to do.
Another note:
I've found this link which is similar to my question but has no answer Proving Riemann integrability using sequences of Riemann sums
Thanks in advance for help!
Best Answer
Given $\varepsilon > 0$, let $\delta > 0$ be such that if $\Pi$ is a partition with $\lambda(\Pi) < \delta$ and $\{t_i\}$ is a sequence of appropriate numbers for $\Pi$ then $|S(f,\Pi,\{t_i\}) - I| < \varepsilon$. By the definition of $\lim_{n \rightarrow \infty} \lambda(\Pi_n) = 0$, there exists $N \in \mathbb{N}$ such that $\lambda(\Pi_n) < \delta$ for all $n \ge N$. Hence $|S(f,\Pi_n,\{t_i^{(n)}\}_{i=1}^{M_n}) - I| < \varepsilon$ for all $n \ge N$. Since $\varepsilon$ was arbitrary, $\lim_{n \rightarrow \infty} S(f,\Pi_n,\{t_i^{(n)}\}_{i=1}^{M_n}) = I$ by the definition of a limit.