Proof tactics, rather than strategy, in real analysis

Question

Frontloading the question portion: What are the most common tactics, as opposed to strategies, used to arrive at viable proofs in (introductory, but also in general possibly?) real analysis? For instance, if I were teaching a standard pre-calculus class I would tell students to pay special attention to the trig identities if they ever plan to take calculus II. What is the equivalent for this?

I generally do not have a problem understanding the high-level strategy of a proof, or at least no more of a problem than the average student does. What I do have a problem grasping is tactics. I've seen this question but it describes what I would call "strategy" here. What I mean by tactics are:

• Knowing, out of nowhere, to use things like the binomial theorem or Bernoulli's inequality or the triangle inequality in a proof. Those are two well-known examples, which I know by now might come up, but I don't trust that there aren't others, and I don't know how to know whatever tactic is necessary to figure out which obscure theorem, probably not taught or dwelled upon in high school (which for me was over 15 years ago) that I need to conjure.

• The whole intervals of epsilon/2 thing, which I don't understand at all, intuitively, mathematically, or rotely.

• One of the early analysis problems was to prove that there exists a real number x such that x^2 = 2. The proof involved conjuring, out of the clear blue sky, the fraction (2-x^2)/(2x+1). I didn't understand where this came from in class -- it's a supremum proof, and most supremum proofs I'd seen seemed to involve things like adding 1 -- or how it is useful, I didn't understand it in any textbook I consulted, and the professor refused to tell me the process by which one comes up with this arcane-seeming and non-obvious fraction. Surely there is a process and someone didn't sit there trying every single fraction they could come up with; I just don't know what it is, and I need to. When something like "prove that for some real x, x^2 = 3" inevitably shows up on the exam, I won't know how to do that either, let alone how to generalize it, which might also show up on the exam. For that matter, I'm not even confident I'd remember this is a supremum problem, since it doesn't look like one.

Answering obvious questions: Our college has an "intro to proofs/pre-analysis" type class, which I did take and do well in, but as far as proof techniques the syllabus covered only the very basics (truth tables, direct/contradiction/contrapositive proofs, etc.) in a way that emphasized high-level strategy rather than tactics. Knowing how to prove that a number is even is all well and good but does not help me with the tactics above. The rest of the class covered things like equivalence classes, (formally defined) functions, countability, etc. -- all of which, while useful, do not help in this regard.

I don't have access to tutoring -- I work 40 hours a week, supplemented with freelancing because I don't actually bill 40 hours a week, a bad work situation that is the entire reason I am in this course, to change careers -- and the tutoring center is 9 to 5. I have tried to talk to the professor, but even when I am able to -- office hours are also generally somewhere within 9 to 5 -- it results in the immensely frustrating experience, which I have now had in three separate courses, to try to prove to a professor that I do not understand the material, have them insist that actually I do, insist myself that no, I am really not even close to understanding the material, have them reiterate that I do, and then fail an exam with something like a 0/100 because, as I tried and failed to express so I could stop this before it happened, I don't understand the material.

Generally if I am told anything it's some platitude like "be creative!" But I am not a particularly creative person, at least not on a time table, and in an exam situation, I don't have time to wait for creativity to strike. Or I am told something like "it took people months to figure this out!" but I do not have months. In an exam situation, I have 90 minutes divided by however many problems are on the exam. I do not have time to "play around." I don't have time to brute-force every arcane mathematical construct -- which, as we learn in this very class, are infinite and maybe not countably infinite. Theoretically I have as much time as I can repeat the course for, but that is bounded by money (I have to pay the full price of the course each time, not to mention being tied to the geographic location of the school and thus the cost of another half-year's rent), by reduced prospects (my graduate school* application already looks terrible because it has multiple W's and a mediocre GPA; the more W's, the worse it looks), and maybe by the fact that the college limits the total amount of courses/credits that non-degree students can take, and I don't know whether withdrawals count.

*Not for mathematics, for teaching math at the grade- and/or secondary-school level, depending on who will accept me.

Answer 1

Here are some tactics I (mostly unconsciously) tried to keep to during these kinds of exams. They are not about being creative. At least $2$ and $5$ are also bad mathematics/life advice, but good advice for introductory analysis/algebra exams.

1) When one possible way of continuing the proof stands out in particular, immediately go with that! Most parts of a proof are "paperwork" and can be done in many different, but actually equivalent ways. Don't waste time pondering on those parts.

2) If a line of thought feels like it doesn't quickly give way, drop it, go back to the last point where you felt "productive" and look for other possible paths. Sometimes, there are one or two key ideas in a proof (most of the time in introductory analysis, there are none), and you should get those right. But you'll have plenty time to try out a few ideas, if you keep to 1) and 2).

3) "Name and conquer". If something seems important, give it a name. If your last line was "The set $S$ is non-empty bounded above by $M$, thus $\sup S < \infty$", then you should give $\sup S$ a name before you continue, e.g. by "Let $m = \sup S$."

4) If there's a definition in your last line, and you don't see an obvious way to proceed, unpack the definition. For example, if $m = \sup S$, we should use that for all $s \in S$, $s \leq m$, and that for all $\varepsilon > 0$, there exists at least one $r_{\varepsilon} \in S$ with $r_{\varepsilon} > m - \varepsilon$.

5) Mathematical structures are built in "layers" on purpose. If a question is to show something holds in groups, then you need to use that an element has an inverse, otherwise it would have been asked for semigroups. If a question asks you to prove that a certain real number exists, then you want to use that every set bounded above has a supremum, otherwise it would have been asked for $\mathbb{Q}$.

6) If an expression looks somewhat similar to one that's occured before, expand it as much as possible and try to isolate the exact difference from the previous expression. Try to find out what can be said about it. This is the whole point in most early examples of proofs by induction, but also very helpful in other situations.

7) Proofs by contradiction are very handy if the case you want to exclude hands you something to work with. Say you want to prove an equation $x=y$, but all you've got so far are inequalities. Then it might be easier to disprove $x < y$ and $x > y$.

Let's prove that there exists $x \in \mathbb{R}$ with $x^2 = 3$, given that $\mathbb{R}$ is an ordered field where every subset that's bounded from above has a supremum (which is how $\mathbb{R}$ was introduced to me).

Now we know that there is no $x \in \mathbb{Q}$ with $x^2 = 3$, hence this isn't a question about ordered fields alone. Therefore, we need a supremum (5), and we need a set for that (4). The nicest thing our supremum could do for us is be $x$ itself, so let's try that (1). How? We need a set that has $\sqrt{3}$ as it's supremum, the simplest possible set that does that is $(-\infty, \sqrt{3})$ (1). This has $\sqrt{3}$ (which we don't yet know exists) in it's definition, but the following has not:

Let $ S = \left\{ x \in \mathbb{R}: x^2 < 3 \right\}$.

We need to check this is really bounded above, but that should be paperwork (because it's "obvious").

If $x>2$, then $x>0$, since $2>0$. Thus $x^2>4>3$, and $x\not\in S$. Therefore, $S$ is bounded above by $2$ and has a supremum. Let $s = \sup S$ (3).

We have what should be $\sqrt{3}$, and we should prove $s^2 = 3$ now. Suprema are all about order, so let's disprove $x^2 < 3$ and $x^2 > 3$, especially since the first expression turns up already in the definition of $S$ (7).

Suppose $s^2 < 3$. Since $s = \sup S$, for all $\varepsilon > 0$, $(s + \varepsilon)^2 = s^2 + 2\varepsilon s + \varepsilon^2 \geq 3$. Therefore, $\varepsilon (2s + \varepsilon) > 3-s^2 > 0$ (6), but since $s \leq 2$, for $\varepsilon < 1$ this implies $5 \varepsilon > 3-s^2$. This is absurd, because $\varepsilon$ can be arbitrarily small, certainly smaller than $\frac{3-s^2}{5}$.

Suppose now $s^2>3$. Since $s = \sup S$, there are arbitrary small $\varepsilon$ such that $s-\varepsilon \in S$, i.e. $(s-\varepsilon)^2 < 3$. Hence, $0 < s^2 - 3 < \varepsilon(2s - \varepsilon)$. But because $1^2 = 1 < 3$, certainly $s\geq 1$ and thus $2s-\varepsilon \geq 1$ for $\varepsilon \leq 1$. This implies $0 < s^2 - 3 < \varepsilon$, which is again a contradiction. Therefore, $x^2 = 3$.

For the last part, we needed $s\geq 1$, which wasn't already proven, but fortunately it's also quite obvious. If a longer claim of this kind turns up while writing a proof, wrap it into a Lemma, which you can prove separately after convincing yourself that it's probably true (much easier than proving it!) and that you can finish the proof with it.

The fraction $\frac{s^2-3}{2s-\varepsilon}$ implicitly turns up very often during this proof (just isolate $\varepsilon$ in each inequality and estimate it by $1$ whenever possible), but it doesn't come out of nowhere, and nobody expects you to be able to "conjure it up". We can tidy everything up now, if there's enough time, and choose particular values for $\varepsilon$ such that everything fits together smoothly, for example a value smaller than $\frac{s^2-3}{2s-1}$ (since we can definitely choose $\varepsilon< 1$).