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I am very familiar with the direct method of proof. Is this proof sufficient enough to be able to understand higher level proofs? Do you need to be fluent in every type of proof? And does knowing how to apply the methods help ease the understanding process of higher maths? I have been self studying maths for a while and I still reach a standstill for long periods of time. I'm wondering if it's due to not being fluent in proof writing.

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The art of the proof, in my opinion, is at the heart of pure mathematics. Eventually, as you do more problems, you will begin to see patterns in your approaches to proof. For instance you might see a problem and immediately think, 'Oh, I should tackle this using induction' or 'this looks like a familiar proof by contradiction'. In essence, by doing more and more math you will build up an intuition for proof that will steer you in the right direction.

Another great (and fun) feature of mathematics is the oft-variety of proofs that can solve a given problem. e.g. One might solve the same integral by integration-by-parts, a u-substitution, a geometric intuition (even/odd), etc..

Do not allow proofs to overwhelm you. With time, you will understand and begin to 'see' more approaches.

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  • $\begingroup$ Excellent advice. Thanks a lot for that. $\endgroup$
    – user606466
    Jan 8 '19 at 17:36
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I'm just going to try to answer this part, based on my own experience of self-learning:

I have been self studying maths for a while and I still reach a standstill for long periods of time. I'm wondering if it's due to not being fluent in proof writing.

It could be due to quite a few things, some of which would be psychological rather than mathematical. And it depends a bit whether the standstill consists of slogging away without making progress—in which case you're slogging away at the wrong things or in the wrong way—or just long breaks from studying any maths.

With self-study it's really tempting to try to go too fast: "Right, I've just about got the idea of that, so let's go on to the next thing". I'd be surprised if you've not done this at all.

But real learning isn't being able to hang on to something long enough to just about get the idea of the next thing, and the next—it's about increasing your understanding.

The difficulty is that it's not always obvious where the gaps in understanding and knowledge are. So here's a suggestion for how to identify them.

First, ask yourself whether what you've just learnt or used feels crystal clear in your mind. If it doesn't, read through it and try to spot the places where you're unsure. Do you actually understand the theorem that was referred to in line $3$? Can you explain why the step from line $4$ to line $5$ is valid? And so on.

You'll probably find a number of little gaps: things you use all the time but aren't quite sure of why they're OK. Things that seem obvious but you can't explain why they're obvious. And those gaps are the things that you need to go back and work on. Don't be surprised if some of them go right the way back to something that was badly explained at school when you were 7 or 8.

Here's an example. Having habitually taken it for granted, I still felt uneasy about using this without justifying it:

if $\sum a_n$ converges on $A$ and $\sum b_n$ converges on $B$, then $\sum (a_n+b_n)$ converges on $A+B$.

So I went ahead and proved that, and found that my proof used the obvious-looking inequality

$$|x+y|\leq|x|+|y|$$

This sort of thing seems so obvious that one just uses it—but with a nagging feeling that "it's obvious" might not be adequate.

So I had to prove the inequality too, which I did like this:

First note that $x\leq |x|$ and that $|-x|=|x|$.

If $a+b\geq 0$, then

$$|a+b|=a+b\leq |a|+b\leq |a|+|b|.$$

If $a+b<0$, then . . . [slightly longer argument because of the minus signs]

. . . So in either case, $|a+b|\leq |a|+|b|.$

Finally, I noted that I'd used

  • $x\leq |x|$
  • $|-x|=|x|$
  • transitivity of $\leq$.

Theoretically I could have gone one step further and spelt out why $x\leq |x|$ and $|-x|=|x|$ follow from the definition of $|\cdot|$. (In the event, I explained it to myself mentally but didn't bother writing it down.)

See what's happened? I've gone right back from proving a theorem about convergent series, to proving basics about simple manipulations—but I've also given myself some practice in constructing proofs, for things which are simple enough to be easily checked.

I feel that this kind of thing is important in self-learning. The point is that without a teacher, you are the teacher, so you have to be able to identify the gaps in your understanding and to check thoroughly whether the logic of a proof is correct. Ultimately this means breaking down the steps until you know that each one is valid.

And I strongly recommend writing an actual list of the theorems and assumptions you've used. Use this list as a guide to what needs consolidating. Don't be satisfied until you can explain the reasoning behind all the list items. When you can, you'll be much more confident in using them. And it's absolutely fine if some of the things you need to explain to yourself are basics. Once they're more solid, the whole structure is more solid.

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