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In one of my proofs, I wrote "Assume, {stating inductive hypothesis}" and in inductive step "We know that ... holds for k-1" instead of writing "Inductive hypothesis: {stating inductive hypothesis}" and "... holds for k-1 from inductive hypothesis". My instructor considers my proof to be wrong/inaccurate because I didn't explicitly state that this is inductive hypothesis. It is not even math or proof course. Would such proof be acceptable?

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    $\begingroup$ I'd say this was up to your instructor. In practice, it's quite common for mathematicians to omit phrases like that when they are clearly understood. But any given instructor may require that proofs be written out in a precise form. $\endgroup$ – lulu Apr 21 at 23:43
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    $\begingroup$ Well, this is a very strange decision from the instructor, that's for sure. $\endgroup$ – Mark Apr 21 at 23:44
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    $\begingroup$ Your proof is acceptable. Those who don't understand mathematics confuse rigurosity with blueprints and made-up rules. "Inductive hypothesis" is just a phrase and can be expressed in many other ways (as you did); there is no wise old man on an island that ordered mathematicians to do so, shall the world crumble otherwise. Don't worry, fellow, you just have a lousy teacher. $\endgroup$ – Lafinur Apr 22 at 0:21
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    $\begingroup$ Your proof is completely legitimate, but as a student sometimes it's just easier to do the little things for your professor. I can't tell you how many times I have been forced to write "thus the inductive hypothesis holds by the principle of mathematical induction" $\endgroup$ – Theo C. Apr 22 at 0:59
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It's not necessary, but it's often enforced in introductory proof-based classes and such to get you used to the flow, and probably to make it easier for professors to address problems regarding such. ... though why you'd be required to do so outside of a mathematical context is beyond me.

That said, explicitly stating this, that, or the other isn't truly necessary. What's important is the underlying logic and process. For example, in my abstract algebra classes, we've done several proofs by induction without explicitly stating the steps.

Of course an argument can be made that explicitly stating the steps can help in terms of clarity, especially for novices. But after seeing ever-so-many induction proofs, you kind of get a "feel" for the overall flow and can see when one is proceeding inductively even if it's not explicitly stated. That's a common theme in mathematics - once there's a sort of common understanding between two people (e.g. reader and author), they might omit the finicky details/notations in favor of brevity and ease of communication.

So, a tl;dr - your proof is completely valid, and your professor is likely enforcing the restriction for moreso their own uses (common errors in the class, making their own life easier, etc.) than your own.

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