The following two is equivalent.
Let $f$ be bounded function from $[a,b]$ to $\Bbb R$
(1) $f$ is Riemann integrable and say $\int_{a}^{b}f=A$
(2) for any $\epsilon \gt 0, \exists \delta \gt 0 $. If partion ||P|| $\lt \delta$, P $\in \Bbb P[a,b]$ where ||P|| is maximum norm of interval in the partition, $\Rightarrow |R(f,P-A)|\lt \epsilon $ (R(f,P) is Riemann sum)
Pf> Proof of My text book. $ \exists P_0=\{x_0,x_1,\ldots,x_n\} \in \Bbb P[a,b]$ such that for $\epsilon \gt 0, U(f,P_0)\lt A+\epsilon$
Now let $M=sup\{|f(x)| : x \in [a,b]\}$b and let $\delta_{1} = \frac{\epsilon}{nM}$. then for any partition P =$\{y_1,y_2,\ldots,y_n\}$such that ||p|| $\lt \delta_1$. Let
$$I=\{i=1,2,\ldots,m : (y_{i-1},y_{i})\cap P_{0} \neq 0\}$$ $$J=\{i=1,2,\ldots,m : (y_{i-1},y_{i})\cap P_{0} =0\}$$
First we consider $f \gt 0$. $$ U(f,P)=\sum_{i=1}^{m}M_i(y_{i}-y_{i-1})$$ $$=\sum_{i \in I}M_{i}(y_{i}-y_{i-1}) + \sum_{i \in J}M_{i}(y_{i}-y_{i-1})$$ $$<nM||P||+U(f,P_0)<\epsilon + A+\epsilon=A+2\epsilon$$
and then my text book said that we can find $\delta_2 \gt 0$ such that if $||P||<\delta_{2} \Rightarrow L(f,P)>A-2\epsilon$ (L(f,P) is lower Riemann sum)
In this point I can't understand how to reduce L(f,P) by dividing index of any partion. could you please give me some explanation?