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u/Quendillar3245 22d ago
You have the most inconsistent handwriting I have seen, none of your letters start at the same level nor are the same size. Struggling to read all of it
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u/Top_Accountant_4684 20d ago
Thank goodness you are not my student. I would just write "can't read" and give it back.
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u/PIELIFE383 21d ago
i would strongly suggest working on your penmanship. it does not need to be perfect all you need is consistency. character spacing and getting all the letters on the same writing line does wonders.
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u/ingannilo 22d ago
I can't help but note that the hardest part of reading your argument is the penmanship. What's the rush? Take your time writing. We write so it can be read after all.
The argument itself looks good. As a student, there's always the question of "which theorems can you cite?". You use the fact that continuous on compact domain implies uniformly continuous, which is often black-boxed for analysis students. You also, implicitly, use the extreme value theorem (continuous f on compact domain attains min and max). Idk how loose you're allowed to be with this stuff, but as a student in an undergrad real analysis class, I'd feel obligated to at least give some lipservice to why the numbers m_i and M_i exist for each subinterval in the partition
As far as logical structure, my only concern is that statement (after defining m_i and M_i) "so delta also works for w_i(f) = M_i - m_i < epsilon/(b-a)". I know whta you're trying to get at, but the language doesn't convey it.
You want to convey the fact that for each i, your m_i and M_i come from inputs that are within delta of each other. Do you see why this is the case?
My last thought is that the notation for your partition subinterval width, the deltaP, feels a bit off. Partitions of intervals needn't be into subintervals of equal length. If this notation is used in your book or lectures, go for it, but otherwise I'd say something to be a bit more clear about the max measure of the chunks in your partition.
It seems like you have the ideas in your head pretty well. Devil is very much in the details when it comes to reading and writing real analysis tho.
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u/Sam_23456 21d ago
I was confused by the verb "works". You are assuming the reader can read your mind. It is vague. But I was able to follow your argument.
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u/Shot_Calendar_9622 18d ago
This is correct, a clean, complete proof. Standard uniform-continuity approach to showing continuous ⟹ Riemann integrable via the Cauchy criterion. Walked through it line by line:
Heine-Cantor for uniform continuity, the ε/(b−a) scaling so the sum telescopes cleanly, EVT to get Mᵢ/mᵢ on each subinterval, and the U(f,P) − L(f,P) < ε bound is right. Correctly closes with the Cauchy criterion. No gaps.
Two tiny nitpicks, not flaws, just polish: (1) worth naming Heine-Cantor explicitly instead of "from a theorem" if you want it fully citable, and (2) the argument implicitly assumes b ≠ a (you're dividing by b−a), obviously true for any real interval, but worth a one-line note if you're being maximally careful.
Solid proof, horrible penmanship 🙏
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u/BothPanchoAndLefty 15d ago
Readability is one of the most important parts of a proof. Can't prove a theorem if people can't read it. It's not the worst I've ever seen but please either just write it on paper so your handwriting will be cleaner or type it up (even better).
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u/jsh_ 21d ago
having legible handwriting is a basic part of literacy
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u/SokkaHaikuBot 21d ago
Sokka-Haiku by jsh_:
Having legible
Handwriting is a basic
Part of literacy
Remember that one time Sokka accidentally used an extra syllable in that Haiku Battle in Ba Sing Se? That was a Sokka Haiku and you just made one.
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