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So I have a soft beginner question about reading proofs. Proof is provided below.

For some reason I may stumble over expressions like this, finding it hard to accept their truth.

The equation $a=qd+r$ yields

\begin{align*} r&= a-qd\\ &= a-q(ak+b\ell)\\ &= a(1-qk)+b(-q\ell) \\ \end{align*} ...and until the end of the paragraph.

I mean I can see from the algebra that $r$ turns out to be of the form sufficient to place it in the set A, but nothing clicks in my mind, there's no intuition for me behind it, and so I find it hard to accept that $r$ is in $A$ (and, for that reason, the rest of the paragraph explaining how $r$ must be $0$), which is what I'm concerned about.

I wonder what should be the subjective feeling as you read through proofs like these? Do you just accept things as true because you accept algebra works (or that axioms, definitions, etc. are true) and you don't question it/them at that moment. You see that algebraic operations lead you to the expression and you accept the end result?

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    Maybe I'm just overthinking... – 2017-02-10

2 Answers 2

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From the division algorithm we know that $r$ is an integer such that $0 \leq r < d$. Right?

Now, given this information about $r$, we also know that we have chosen $d$ to be the smallest positive element in the set $A$.

So, once we (manage to) show that $r$ is also in $A$, then from the information we have at our disposal we can conclude that $r$ cannot be positive and so must be zero.

Thus everything boils down to showing that $r$ is in $A$. For this, we solve the division algorithm equation for $r$, and finally substitute the expression for $d$ and then collect terms, thereby arriving at our desired form for "placing" $r$ into set $A$.

Does this help you?

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    I understand the proof, it's a little different question. It's just that it seems that we've established this fact that r is in A somewhat arbitrarily through operations that we know to be true, so it's almost like this happened by chance and we must accept it because we know everything we did was done correctly. – 2017-02-10
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    I guess the question was do you just accept the logic, even if there's no intuition behind what you're proving? What's your take on it? – 2017-02-10
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My subjective feeling in the case of stumbling would be that the proof, though perfectly correct, is not well-written. Well-written proof should be designed in such a way that the reader could understand ideas behind it and maybe understand how it was possible to come up with such a proof. In this particular case the proof failed to do this: it is written in a way that it is difficult to imagine how it was constructed. However, in general it is not always the case: it may as well mean that the statement is just tough to comprehend at your current level in math.

Of course, you shouldn't question the validity of a proof if you checked reasoning. But if you don't understand the deep logic behind some proof, consider googling the proof of the same statement in another place. However, note that math is hard and it is common that almost every proof in some book calls the same reaction. This may point out that you use a wrong book.

Anyway, if the topic is hard it is usually not a big problem if you don't understand ideas while understanding reasoning, as soon as you manage to do the provided exercises on the topic.

P.S. I'm an undergraduate student who has some experience in math and reading math books. So my feelings can be wrong, and probably you can get a much better answer on the topic from people who teach math.