# Grade on proving |$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$

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

• 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. Mar 31 at 11:15
• 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.
Mar 31 at 13:36
• 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. Mar 31 at 17:16
• 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. Mar 31 at 17:55
• Did the professor give you a rubric for grading? Mar 31 at 21:51

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 $$n$$".
• @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." Apr 4 at 23:02