Knowing a proof versus understanding it

Question

How does one make the jump from merely "knowing" the steps required to prove something to being able to deeply understand them? I can often look at a proof/see a professor explaining it and I am convinced that the proof works, but I don't understand it well enough so that I would have been able to come up with a similar idea myself had I not seen the proof. Often the proof just "happens to work" and I'm not able to see why it is the best way (or one of the best ways) to prove the assertion.

How does one make the jump from knowing the steps in a proof (and getting why they work) to truly understanding the ins and outs of it?

Edit: A better idea of what I mean by understanding - Say you come up with a basic idea (a very rough plan to prove something) but you aren't able to formulate the argument fully; then you see the a proof in a book and it essentially takes your idea and proves the thing rigorously. How does one understand the proof well enough so that one would have been able to see the idea themselves had they, hypothetically, forgotten the proof but somehow retained the "understanding" (and lost the "knowledge")?

An example. Let's take a proof of a very basic claim in real analysis: every compact set in $\mathbb{R}^{n}$ (in fact in any Hausdorff space) is closed.

Let $K \subset \mathbb{R}^{n}$ be compact. We want to show $\mathbb{R}^{n} \setminus K$ is open. Pick $x \in \mathbb{R}^{n} \setminus K$. It suffices to show that $B(x, \varepsilon) \subset \mathbb{R}^{n} \setminus K$ for some $\varepsilon > 0$. Then, for all $k \in K$, there is $\varepsilon_{k} > 0$ and $\varepsilon'_{k} > 0$ such that $$ B(x, \varepsilon'_{k}) \cap B(k, \varepsilon_{k}) = \emptyset $$ The collection $\{B(k, \varepsilon_{k})\}_{k}$ is an open cover of $K$ and we can find a finite subcover $\{B(k_{i}, \varepsilon_{k_{i}})\}_{i=1}^{N}$. Next, we can pick $$ \varepsilon= \min_{1 \leq i \leq k} \{\varepsilon'_{k_{i}}\} $$ and $B(x, \varepsilon) \cap K = \emptyset$, so we have shown that $K$ is closed.

Now, this isn't a super hard proof and only makes use of basic facts, but seeing this is a beginner can be hard (and was for me when I first saw it). I got what we were doing, but I didn't see deeper. But somehow while reproducing this proof right now, the idea of "separating the set and the outside using balls" was intuitive to me: what made me jump from "knowing" to "seeing/understanding" what had to be done?

Answer 1

I agree with Omnomnomnom's comment. However, I think that the generic question (i.e. not tied to a specific math problem) is so outstanding that I will try to provide an example.

Suppose that you are asked to prove that $\;|a + b| \leq |a| + |b|.$
The first thing to do is to metacheat:
a. Assume that the hypothesis is true.
b. Assume that there is a reasonably straight-forward way of proving it.
c. Assume that the proof entails the concepts that you have recently been studying.
Otherwise, what is the point of presenting this problem at this time?

Second, look at examples:
a. $a=5, b=3.$
b. $a=-5, b=-3.$
c. $a=5, b=0.$
d. $a=5, b=-3.$
e. $a=-5, b=3.$
f. $a=-3, b=5.$

Third, look for a pattern:
If $a$ is positive and $b$ is negative, or vice-versa,
then the LHS ($|a+b|$) is less than the RHS.
Otherwise, you have equality.

Fourth, you are still not ready to attempt a proof.
Try to visualize why the pattern holds.
If you consider $a$ and $b$ vectors,
and you construe $|a| + |b|$ as the total distance traveled
then you might reasonably construe $|a+b|$ as the resulting distance from the origin.

In this construance, it is intuitively reasonable that the LHS < the RHS when $a$ and $b$ have different signs.

This is because the different signs cause a change in direction.

This is the Oh! moment, where you have stretched your intuition.

Now, if you try to algebraically prove the hypothesis, everything should fall into place.

Obviously, this approach is crafted for this particular problem. However, this approach may serve as a guide for other math proofs.

Answer 2

I don't understand it well enough so that I would have been able to come up with a similar idea myself had I not seen the proof.

I don't think that's a good criterion for understanding a proof. Math makes progress over many generations, and we're fully allowed to use the cleverness of those who came before us. Very few mathematicians would have come up with, say, the Prime Number Theorem on their own - but they are still able to understand and appreciate the proof.

I think better criteria for understanding a proof are

• Being able to reproduce the proof days later (without rote memorization).
• Understanding why each of the hypotheses is required. That is, for each hypothesis, either come up with a counterexample of the theorem failing when that hypothesis is missing, or be able to generalize the proof to avoid or weaken that hypothesis. For example, can you generalize the theorem you gave to spaces beyond $\mathbb{R}^n$? If coming up with counterexample or generalizing is difficult, you should at least understand where in the proof each hypothesis was used and why it becomes more difficult without it.
• Being able to use the general technique of a proof in a different problem. For example, the proof that a continuous function on a compact set is uniformly continuous uses a similar method: cover the domain with small $\epsilon$-balls, use compactness to get a finite subcover, and then take the smallest $\epsilon$. Having seen this method before, you won't be so surprised when it comes up again.

So: Don't beat yourself up too much if you wouldn't have been able to come up with a proof yourself. Instead, just use the knowledge you've gained from reading that proof to see what else you can do with it.

As for your main question:

How does one make the jump from knowing the steps in a proof (and getting why they work) to truly understanding the ins and outs of it?

• Do problems - lots of them.
• Take a deep breath. Stand up and take a walk.
• If you can't understand the proof now, it's okay. Come back later, if you have time, and read the proof again. Each time you do, you'll understand it a bit better.
• Accept that some proofs are hard and you might not understand them. That's fine too. No one has time to understand every theorem on earth. (It's okay to occasionally use a theorem as a "black box", just taking the statement of the theorem as a given.)

Answer 3

One good way to develop this sort of intuition is to spend lots of time searching for the proof prior to reading it - and, perhaps even better, to spend time coming up with conjectures prior to reading theorem statements. This doesn't always lead to a result, but it gives you practice in exactly the skillset you need. It's also often worth reading a proof in a book and thinking, "Oh, I don't get this at all" - and then trying to find your own proof that makes more sense. Either, you'll find that the proof in the book was terrible (which happens a lot) or you'll find yourself forced into the path that the book took - and will hopefully understand the proof better after finding it unavoidable. It's also good to look at bigger theorems and ask yourself how you would prove it from the ground up - because you'll often find that proofs fit together in nice ways.

If you want to prove:

Every compact set $K$ in $\mathbb R^n$ is closed.

You should probably immediately expand this to the start of a proof:

Let $x \in \mathbb R^n\setminus K$. We wish to find some $\varepsilon > 0$ such that $B(x,\varepsilon)\cap K=\emptyset$ given that $K$ is compact.

Then, there's a bunch of ways to go depending on what you know about compact sets, but going with the most literal interpretation, we know that we're supposed to find some open cover of $K$ and then take a finite subcover, but we don't know how that could help us.

However, we might get a hint by asking how this could fail: we would have a problem if there was, for every $\varepsilon > 0$ some $y\in K\cap B(x,\varepsilon)$. Maybe if you have a good mental picture of compactness, your mind will make the leap right here - but if not, you can surely draw some potential counterexamples - just draw a bunch of concentric circles around a point $x$, and put at least one point $y$ in each circle, and ask yourself why this is not compact.

You can start with the simplest example, which would be a sequence of values $y$ converging to $x$ in some controlled manner (e.g. $\{1,1/2,1/3,1/4,1/5,\ldots\}$ with $x=0$) and figure out why this is not compact. You might find this hard the first time you try it - if you've never seen an example of non-compactness, you might need to be creative to figure out which open cover lacks a finite subcover. But, this is certainly a more manageable question than you started with - "show this set isn't compact" is way easier than "show all compact sets are closed."

Hopefully, you'll eventually see why this set isn't compact. If your counterexample to compactness isn't already written in terms of the balls $B(x,\varepsilon)$ that you initially started with, see if you can make it so that it does - you want this to generalize, and the only things you've really got so far is $x$, the balls around it, and, if we're thinking about counterexamples, some $y$. Okay, so maybe you don't see how to do that - maybe your brain is now stuck in "we got a sequence converging to something." Think of some other "counterexample" then - find some other set that's not closed and think about why it's not compact. Maybe you think about a disc minus a single point or a spiral closing in on your given point. Try to see non-compactness solely with regards to some balls around a point.

After enough experimentation, one would hope that you'll come up with the thought that every point in $K$ is some positive distance from $x$ - so is outside of $B(x,\varepsilon)$ for some $\varepsilon$. Then you're at least thinking about the sets $\mathbb R^n\setminus B(x,\varepsilon)$ - and, since you still have the overall goal of finding some open cover of $K$ to talk about, it should occur to you that these complements are closed - but maybe you can just modify them to be open and consider $\{y\in K: d(x,y) > \varepsilon\}$ or equivalently $\mathbb R^n\setminus \bar B(x,\varepsilon)$. You don't need to see the end yet, but, hey, you got an open cover of a set that seems to have something to do with the your overall goal - that's progress. Might as well see what using compactness yields.

Then, again, you hit some place where your mind could just leap to the answer, but if it doesn't, you can go back to the concrete examples you were thinking about and maybe modify them to take out some ball around $x$ from $K$ so that they become examples instead of counterexamples. What would compactness give then? How would we find the ball around $x$ we require just given that finite subcover? Hopefully, you'll stumble on the idea that, since the sets in your cover get larger and larger as $\varepsilon$ decreases, a finite subcover is going to have one element that contains all the others - so you would have just proven that $d(x,y) > \varepsilon$ for all $y$ and some fixed $\varepsilon$. Okay, now you're done. Go back and figure out the path you took, and write it down as a proof - and it's worth taking the time to write the proof down clearly (which is a whole can of worms by itself - but let's not get into proof writing).

Of course, you could be done then, but maybe you want to look for other proofs of the same fact, or maybe you want to find the most efficient way to develop real analysis. Maybe you come back some later day where you're more familiar with the topic as a whole and look at your proof to see if it has any commonalities with other proofs that might be pulled out as lemmas. You might recognize that the quantity $d(x,y)$ is appearing a lot - and hopefully, at some point before you stop studying the subject, you'll see that $f(y)=d(x,y)$ is a continuous function of $y$ - and really, your proof is just trying to show a positive lower bound for $f$ on $K$. You might too realize that your sets are just of the form $\{y\in K : f(y) > \varepsilon\}$, which might be reminiscent of the extreme value theorem's proof - and, oh hey, if we applied the extreme value theorem to this $f$ - which we know to always be positive - what do we get? Oh look - you just discovered that your proof of the original theorem was really just a little lemma on continuity of the distance function smashed together with the proof of the extreme value theorem - and, hey, now you have a name to give to all that fiddly business with open sets and balls, which should help you see the bigger picture even better than you would already have. Cool.

There's a ton of little intuitions and skills that I list above - I'm asking you to expand definitions and theorems without any trouble, to look at the contrapositives and negations of theorems you're interested in (and of the statements that arise in a proof), to keep clear both overarching goals of a proof and more minute ones, to instantiate broad statements down into specific examples, and, generally, to be perseverant in your work. These are skills you only really gain when you sit down with something hard and push yourself to produce something. Even when the path to your ultimate goal isn't clear, you are rarely ever truly without anything to do - if you are stuck, you should find something that you can do and which looks relevant and work on it - the worst case scenario is that you either find something simpler that you also don't understand (which is great - work on figuring that out first!) or you end up somewhere weird (which can also be great - there's lots of interesting mathematics that textbooks don't mention, but that you might run into exploring on your own). You'll get better at all the little skills doing this, and sometimes, you'll even make it all the way past the obstacles and come up with big proofs on your own - which is the surest way to understand a proof.

In short: it sounds like your goal is to go from knowing proofs to feeling able to produce proofs. The way to do that is to practice producing proofs.