I recently completed a solid chunk of my first truly formal math book on Set Theory and decided to move on to Michael Spivak's Calculus (as I heard it was a good place to begin one's journey into Analysis). For the purpose of self-studying, I purchased the solution manual, as well.

After completing the first two chapters and going through all practice problems, I am now painfully aware that I am not particularly great at proofs involving clever equation manipulations. As I progressed through my aforementioned set theory book, I felt like the number of "proof flavors" (different strategies one could invoke to tackle the claim) were quite manageable. Figuring out which path to pursue seemed to be categorically recognizable.

However, when it comes to proofs involving equation manipulation, I literally feel like I am staring into a sea of hundreds of possible permutations (in the English sense) that one might use to arrive at the desired conclusion. From what I have seen so far, the strategies largely seem to involve the invocation of Field Axioms (e.g. creative usages of $0$ and $1$ through the existence of additive inverses and multiplicative inverses, respectively) and factoring (that often times feels quite contrived).

I was wondering if there were any general tips and tricks as to how one might improve his/her ability in recognizing the type of equation manipulation that a given proof is suggesting (yelling) to carry out. I am assuming the seasoned mathsters have some sort of pattern recognition that goes off in their heads that allows for efficient determination of starting points.

I apologize if this question is a little too broad. I recognize that every proof may necessarily be "special", so in the absence of a specific example, perhaps this question is unanswerable. But here's hoping. Book recommendations are also greatly appreciated! Cheers~

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    $\begingroup$ Typical proofs in analysis do not involve much algebraic manipulation. Instead they deal with basic inequalities (ok mostly triangle inequality). $\endgroup$ – Paramanand Singh Feb 27 at 3:06
  • $\begingroup$ Can you provide some specific examples from Spivak where you think algebraic manipulation part of some proof is tricky? $\endgroup$ – Paramanand Singh Feb 27 at 3:07
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    $\begingroup$ You have to crawl before you can walk. The reason that you are having trouble is probably nothing more than lack of experience. Instead of jumping into Spivak, find an intermediate book with lots of problems, and get good at that intermediate book. Then move on to Spivak. $\endgroup$ – user2661923 Feb 27 at 3:25
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    $\begingroup$ In general algebraic manipulation is mostly a thing of practice. If you have seen/done many examples you will be able to do more. $\endgroup$ – Paramanand Singh Feb 27 at 4:02
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    $\begingroup$ The way one finds "clever" proofs is to immerse yourself deeply into the problem and live in it for so long that you have explored all the ins and outs and eventually stumble into the clever solution. Then you write it up nicely and pass it around and everyone is amazed how you so easily found this neat proof. They don't see all the blood, sweat and tears, and mostly time, you put into it. You've only begun to put that effort in. Don't get discouraged because you cannot yet match those who have invested far more effort. $\endgroup$ – Paul Sinclair Feb 27 at 14:38

I'm about halfway through Spivak's Calculus myself.

The specific problem you mention (Prove $\sqrt 6-\sqrt 2 - \sqrt3$ is irrational) is a tricky one. I certainly wasn't able to do it myself. I ended up having to look at the published solution. Coming back to it today, several months later though, I managed it.

(Note: the next part of that problem ("prove that $\sqrt2$ + $\sqrt[3]{2}$ is irrational") is marred by a hint ("start by working out the first 6 powers of this number") that's meant to push you towards the published solution, which relies on linear algebra and is certainly outside the scope the book. Usually Spivak is loath to rely on things without justification. This is an unusual lapse, made more baffling by the existence of a much more appropriate solution. Compounding things, I think there's also a typo in the table in his solution. There are mistakes in the text, ranging from typos to more serious problems. When you think there might be a mistake, MSE can be very helpful.)

More generally, many of the problems in the book are challenging. Often, I have to look up the solution, at least for a hint.

The difficulty is also somewhat front-loaded. The first 4 Chapters which in 3rd edition are "Basic Properties of Numbers", "Numbers of Various Sorts", "Functions", and "Graphs" (with 3 appendices), were in some ways more challenging for me than anything that's come since.

Once you hit the first chapter on "Limits", things settle into a groove.

Not that it's "easy" from that point on; the problems remain challenging throughout. Just that, I think you'll find from limits on, it gets really fun.

Don't be discouraged if you aren't able to "ace" the problems. Many of the early problems are introducing you to concepts and techniques you will use later on. The full import of some things doesn't become clear until many chapters later.

You can learn most of Spivak's clever tricks in Spivak, as opposed to trying to learn them elsewhere ahead of time just in case they are in Spivak.

Certainly look outside of Spivak if you need to brush up on something specific, but I don't know if it's worth it trying to just generally get better at clever arithematic tricks before moving on with the text.

Remain diligent about doing the problems. If you make sure you understand them, even the ones you need help with, you will get a lot out of it.

Go back periodically and try re-working problems that initially gave you a hard time.

The book is a tough, glorious, frustrating, edifying beast. Stick with it at least until the end of the "Limits" chapter. I think at that point you'll be happy you did.


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