# Thread: Starting point for Riemann sum, Integral proof?

1. ## Starting point for Riemann sum, Integral proof?

Suppose f is integrable on the interval [a,b] and Q is a partition of [a,b]. Suppose $\epsilon >0$. Prove that there exists $\delta >0$ such that if P is any partition of [a,b] with $|P|<\delta$ and S(P,f) is a Riemann sum then

$|S(P,f)-\int_a^bf|<\epsilon$

I'm mostly just looking for a starting off point, or a direction to go. Should I try proving that $S(P,f)=\int_a^bf$ for $\delta$ small enough? How should I define $\delta$?

2. Still looking for advice. Anyone?

3. Originally Posted by paupsers
Suppose f is bounded on the interval [a,b] and Q is a partition of [a,b]. Suppose $\epsilon >0$. Prove that there exists $\delta >0$ such that if P is any partition of [a,b] with $|P|<\delta$ and S(P,f) is a Riemann sum then

$|S(P,f)-\int_a^bf|<\epsilon$

I'm mostly just looking for a starting off point, or a direction to go. Should I try proving that $S(P,f)=\int_a^bf$ for $\delta$ small enough? How should I define $\delta$?
what does $|P|$ mean? Is it the mesh?

4. Yes.

5. Still trying to figure this one out. Anyone have any ideas?

6. Originally Posted by paupsers
Suppose f is bounded on the interval [a,b] and Q is a partition of [a,b]. Suppose $\epsilon >0$. Prove that there exists $\delta >0$ such that if P is any partition of [a,b] with $|P|<\delta$ and S(P,f) is a Riemann sum then

$|S(P,f)-\int_a^bf|<\epsilon$

I'm mostly just looking for a starting off point, or a direction to go. Should I try proving that $S(P,f)=\int_a^bf$ for $\delta$ small enough? How should I define $\delta$?
As stated the statement is false. That is why on one has answered.
It seems to say that if a function is bounded on a closed interval it is Riemann integrable on the interval.
That is clearly not true.

On the other hand, if you mean that the function is bounded and Riemann integrable then the conclusion is the very definition of Riemann integrable. So there is nothing to prove, unless you are using a different definition. Are you?
If so, what is it?

I suggest that you review the posting and make adjustments.

7. Yes, you're right, part of the hypothesis should be that the function is Riemann integrable.

Does this proof work?

Proof: We know $L(P,f)\leq S(P,f)\leq U(P,f)$
We also know that $L(P,f)\leq \int_a^bf\leq U(P,f)$
This implies that $L(P,f)\leq |S(P,f)-\int_a^bf|\leq U(P,f)$
Since f is integrable on [a,b], we know there exists a partition P such that $U(P,f)-L(P,f)<\epsilon$.
This implies that $|S(P,f)-\int_a^bf|<\epsilon$. QED.

8. Originally Posted by paupsers
Yes, you're right, part of the hypothesis should be that the function is Riemann integrable.

Does this proof work?

Proof: We know $L(P,f)\leq S(P,f)\leq U(P,f)$
We also know that $L(P,f)\leq \int_a^bf\leq U(P,f)$
This implies that $L(P,f)\leq |S(P,f)-\int_a^bf|\leq U(P,f)$
Since f is integrable on [a,b], we know there exists a partition P such that $U(P,f)-L(P,f)<\epsilon$.
This implies that $|S(P,f)-\int_a^bf|<\epsilon$. QED.

I think the problem for me is I don't know what the hell $S(P,f)$ is. But, assuming that $L(P,f)\leqslant S(P,f)\leqslant U(P,f)$....since $f$ is integrable $L(P,f)\leqslant \int_a^b\text{ }f\text{ }dx\leqslant U(P,f)$. So, $\int_a^b\text{ }f\text{ }dx-S(P,f)\leqslant U(P,f)-L(P,f)$ and equally valid we see that $S(P,f)-\int_a^b\text{ }f\text{ }dx\leqslant U(P,f)-L(P,f)$. Combining these we get $\left|\int_a^b\text{ }f\text{ }dx-S(P,f)\right|\leqslant U(P,f)-L(P,f)$ for every partition $P$. Thus, since $f$ is Riemann integrable we see that for every $\varepsilon>0$ there exists some $P\in\wp$ such that $\varepsilon>U(P,f)-L(P,f)<\left|\int_a^b\text{ }f\text{ }dx-S(P,f)\right|$ and teh conclusion follows.
We know $L(P,f)\leq S(P,f)\leq U(P,f)$
We also know that $L(P,f)\leq \int_a^bf\leq U(P,f)$
This implies that $\color{red}L(P,f)\leq |S(P,f)-\int_a^bf|\leq U(P,f)$
This implies that $|S(P,f)-\int_a^bf|<\epsilon$. QED.
But we can say $S(P,f)\leq U(P,f)$ and $-\int_a^bf\leq -L(P,f)$.
Then $0\le S(P,f)-\int_a^bf\leq U(P,f)-L(P,f)$.