"Magical" number and equations in principle of mathematical analysis I am currently a freshman in Math major and I'm trying to study Rudin's principle of mathematical analysis during the summer break, but I'm confused about a few things and hope you guys can help me out.
When I was working on the exercises of Chapter 2 and 3, I can always    find some interesting numbers and designs of equations which will    solve the whole proof with relative ease. Sometimes, the logic of the proof makes sense, but how to design those magical equations are    confusing.
For example Exercise1
In this case, delta = min(sqrt(2-x^2 / 3), 2-x^2 / (3*abs(x)), in the first paragraph of solution . The design of delta forms a nice and easy inequality in the later proof.
Another example would be:Exercise2
I was focusing on finding the limit of a(n) when n approaches    infinity, and see if it is zero, but the answer key shows such an easy way to solve this. To be honest, I have no clue about how to get such equations, but I love them.
I assume I would meet this type of numbers and equations in the    future as well and I hope I could have the same ability to design    such equations to prove stuff. Is there a specific way to train this    ability? Or is it something that I will be capable of when I was a    senior?
Any help and advice would be much appreciated and thanks in advance
 A: 
Is there a specific way to train this ability?

Many would argue that this is what it means to be a mathematician. Most would agree that it constitutes a lot of what mathematicians do.
What I am talking about is the power of constructing your own definitions and proofs. There is no better way to do this than study examples -- especially non-examples, which fail to satisfy some condition of a definition, theorem, etc.; and counterexamples, which show that a statement is false. Another way is to try to design your own examples, usually going the other way as suggested in an exercise (which is usually harder than the routine or suggested way). Then, always try to prove statements by yourself first -- you'll gain much more this way, whether or not you eventually succeed in finding a proof.
Of course, all this will be initially hard as math education at the lower undergraduate level is more of training and exercising than researching and solving problems (and a lot can be said for this, pedagogically), but gradually you will be much more built for what it is mathematicians do and get better at doing these things, and finally at understanding and enjoying mathematics. Good luck.
