Thread: Peculiar means of proving a theorem on integrability

1. Peculiar means of proving a theorem on integrability

This theorem has probably been seen a dozen times here, and I could find other formats of proving this theorem online, but the issue I'm having right now is a context/format issue. The question, listed below, is very specific on how we're supposed to answer it, and I'm not entirely sure how I'm supposed to format it.

If $f$ is integrable, so is the absolute-value function $|f(x)|$. Prove this by showing that if $s,S$ refer to $f$, and $s',S'$ refer to $|f|$, then $S'-s'\leq S-s$. Then use the following theorem.
(for this, $s$ is a lower sum, and $S$ is an upper sum)

"Suppose $f$ is bounded on $[a,b]$, and suppose that corresponding to each positive $\epsilon$ there is a partition of $[a,b]$ such that the corresponding upper and lower sums satisfy the inequality $S-s<\epsilon$. Then $f$ is integrable."

I'm guessing that to solve this one, I'd first need to solve this earlier question: click here.

2. Originally Posted by Runty
This theorem has probably been seen a dozen times here, and I could find other formats of proving this theorem online, but the issue I'm having right now is a context/format issue. The question, listed below, is very specific on how we're supposed to answer it, and I'm not entirely sure how I'm supposed to format it.

If $f$ is integrable, so is the absolute-value function $|f(x)|$. Prove this by showing that if $s,S$ refer to $f$, and $s',S'$ refer to $|f|$, then $S'-s'\leq S-s$. Then use the following theorem.
(for this, $s$ is a lower sum, and $S$ is an upper sum)

"Suppose $f$ is bounded on $[a,b]$, and suppose that corresponding to each positive $\epsilon$ there is a partition of $[a,b]$ such that the corresponding upper and lower sums satisfy the inequality $S-s<\epsilon$. Then $f$ is integrable."

I'm guessing that to solve this one, I'd first need to solve this earlier question: click here.

I think this follows almost at once from $\left||a|-|b|\right|\leq |a-b|\,,\,\forall a,b\in\mathbb{R}$

Tonio

3. That's the reverse triangle inequality. It could definitely apply here, though in what way is the tricky part.

Just so you know, this question came out of Taylor & Mann Advanced Calculus, 3rd Edition (if you have this book, look at Page 539). And as I said, this question seems to have a VERY particular means in which it must be solved. Trying to answer the question in this particular way is proving quite difficult.

I'm not sure if this might help in properly answering the question, but this lemma from the textbook may be relevant to the subject at hand.

Suppose we start with a certain partition, and then obtain a new partition by inserting some additional points. Let $s,S$ refer to the lower and upper sums for the original partition, while $s',S'$ are the sums for the new partition. Then $s\leq s'$ and $S'\leq S$.

I don't know if this lemma would help at all, but it's from the same chapter as the question. Maybe it has some worth.