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In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$$ for $n$ real numbers $a_1,a_2,...a_n$

I am teaching assistant for this course, and one of my students replied like this:

$|a_1+a_2|\le |a_1|+|a_2|$ by triangular inequality.

Then, $|a_1+(a_2+...+a_n)| \le |a_1|+|a_2+...+a_n|$

$|a_1|+|a_2+(a_3+...+a_n)| \le |a_1|+|a_2|+ |a_3+...+a_n|$

Repeating this ,

|$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$

I gave him a bad grade (5 from 10), so he complained. It is my turn to reply.

  1. Is the student's proof correct?
  2. Is the point deduction justified? The professor and I expected the students to use mathematical induction for this proof.
  3. How can I most clearly communicate the grading rationale to the student in our upcoming conversation? How can I defend the grading policy against student objections?

I posted the question on mathstackexchange too: https://math.stackexchange.com/questions/4083896/grade-on-proving-a-1-a-2-a-n-le-a-1a-2-a-n?noredirect=1#4083901

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    $\begingroup$ Whether mathematical induction should be used to prove the result at this specific place in this specific course depends on expectations for proofs in the course, and thus is not something we can competently answer. And if mathematical induction should have been used, the amount taken off should be proportionate with the perceived importance of using mathematical induction at this time in this course --- some would argue that the use of mathematical induction is obvious and thus not the main idea, but maybe this part of the course is still training students to use mathematical induction. $\endgroup$
    – Dave L Renfro
    Mar 31, 2021 at 11:15
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    $\begingroup$ I think that this is a completely valid proof and I would have given it full marks. As to whether it is inductive, I'd argue that the style is more recursive than inductive but that this is more about fitting the proof into a template rather than any of the actual content. $\endgroup$
    – Adam
    Mar 31, 2021 at 13:36
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    $\begingroup$ This proof is correct and deserves full credit unless the students have been taught (and are expected to demonstrate here that they have learned) that "repeating" (and synonyms like "continuing") are abbreviations for rigorous inductive proofs. $\endgroup$
    – Andreas Blass
    Mar 31, 2021 at 17:16
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    $\begingroup$ Agree with above comments. Additionally, I will say that the arbitrariness of the level of rigor demanded of a student can feel malicious, and definitely turn students away from thinking of mathematics as a "reasonable" field. $\endgroup$
    – Steven Gubkin
    Mar 31, 2021 at 17:55
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    $\begingroup$ Did the professor give you a rubric for grading? $\endgroup$
    – Sue VanHattum ♦
    Mar 31, 2021 at 21:51

1 Answer 1

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In my opinion whether this proof is correct or not depends on the actual wording of the question. This recursive argument shows that the statement is true for 100 numbers (since it is true for 2, 3, 4, so we eventually reach 100) It also shows that the statement is true for 1000 numbers (we eventually reach 1000). For any fixed natural number this is a correct argument. However, if the students were asked to show that the statement is true for all positive integer $n$, then an extra step is needed. This is where quoting the principle of mathematical induction comes in. It is important for students to understand that the principle of mathematical induction is not just a shortcut to shorten a proof. It is a necessary axiom to move from "true for any fixed $n$" to "true for all $n$".

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    $\begingroup$ Can you give an example of a statement that is at the level of an advanced calculus student that is true for any fixed n but not for all n? This explanation is citing a very subtle distinction. $\endgroup$
    – Steve
    Apr 2, 2021 at 17:26
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    $\begingroup$ @Steve The distinction is subtle but real, at least to people interested in foundations. The student solution follows the idea of presenting an algorithm to verify the validity of a statement for any given finite set. This settles nicely with finitism. But if we acknowledge the existence of an infinite set N as in ZF and want to prove the statement hold for the infinitely many cases in N, then we have to bring out the axiom of infinity. Some answers here clarify: math.stackexchange.com/questions/2850064/… $\endgroup$
    – user52817
    Apr 2, 2021 at 21:58
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    $\begingroup$ @user52817 It's not that /I/ don't believe you, but part of the purpose of this question was to solicit a plausible explanation or defense that the student can at least comprehend even if they disagree. I don't think "the quantifier is in the metatheory, which doesn't always translate to a quantifier in the formal theory, and that is important not for any reason I can articulate to you but because it is important to some remote field to which you have not yet been exposed and likely never will" meets that threshold. $\endgroup$
    – Steve
    Apr 2, 2021 at 22:34
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    $\begingroup$ @Steve I think we are in agreement here. I am suggesting that we as educators delude ourselves when we think students in calculus are developmentally ready for this foundational issue. Requiring students to formally invoke and chant the axiom of infinity or principle of induction is unnecessary, unless we are specifically trying to teach the foundational issue. $\endgroup$
    – user52817
    Apr 2, 2021 at 22:56
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    $\begingroup$ @steve It is the statement itself that makes the difference. Here is an example. "For 100 real numbers, the magnitude of the sum is at most the sum of the magnitudes." This is one statement and it has a long proof (when we replace the "repeating this" of the original post with the appropriate steps. The proof is similar (but longer) when we change 100 to 1000, or any number. These are infinitely many statements with similar (but different) proofs. Some extra tool is needed for proving these at once "For any positive integer $n$, the magnitude of the sum is at most the sum of the magnitudes." $\endgroup$
    – Ferenc Beleznay
    Apr 4, 2021 at 23:02

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