For last couple of weeks I am trying to read "Game Theory" from Fudenberg and Tirole and "Microeconomics" from Jehle and Reny. But the kind of maths they use for their theorem-proof-theorem analysis, is just not getting into my head. What book can I use to easily master such kind of Real Analysis that can help me to understand these books?
If you don't have any background in Real Analysis, I suggest you to read "Analysis I" written by Terence Tao. I was in a similar situation that you a year ago. I studied Economics at college but we never had to do rigorous proofs (as a matter of fact, I did not know what was a theorem, a mathematical proof, or mathematical logic until I read the book). We only learned some "rules" of derivation and integration, and to manipulate algebraic equations in order to get some important results in Economics. However, a professor suggested me to apply to a PhD program and to take a real analysis course. It was really hard because I had never seen something like that. The books recommended by the analysis professor were the common books you will be recommended here: Baby Rudin, Kolmogorov and Fomin, Ok, Abbot, Apostol, etc. I read some of them (specially Baby Rudin) and I found them too difficult.
The problem was that I found difficult "thinking as a mathematician". Therefore, I started to search for the "best book" in real analysis, and I found this one written by Terence Tao. It was amazing, not only because the deep understanding and the outstanding expository skills of the author, but because he started from the very beginning: he answers some important questions (what is analysis and why to do it?) and starts explaining what are the natural, integer, rational, and real numbers (and also the Peano axioms, and some operations). He also starts explaining what sets are. Then, he develops the most important concepts of analysis: sequences, limits, convergence, series, etc.
I think the most important feature of the book is that it is completely focused on building mathematical skills: how to prove a proposition, different ways to do it, how to use your intuition to develop a proof, etc. And also, it gives you a complete understanding of the topic. This makes it starkly different from other books such as Baby Rudin (which assumes that you know lots of results that, in fact, you don't), which is useful as a reference book and also when you have some well understanding of the topic (I moved easily from Tao to Rudin when the analysis course I took required me to do so). Another important characteristic of the book is its appendix: it gently introduces you to the basics of mathematical logic: what mathematical statements are, the structure of proofs, (nested) quantifiers, etc; and gives several examples of proofs.
I often see questions like this and I have always found puzzling why no one recommends this book. It is maybe the best mathematics book I have ever read (as it may seem obvious, I love that book I have fond memories of me learning how to make proofs and what real numbers and analysis really are). Also, it is strange because Terence Tao is unarguably the most important mathematician alive and a very kind person. You can see his blog and the books he has published in the following link: https://terrytao.wordpress.com/books/. I hope this recommendation would be useful.
It's hard to make a best advisement given that we don't know your background. One can struggle with proofs for a few reasons: terseness of the presentation, a weak background in the prerequisite material, or a weak background in proof-writing in general. (1) and (3) are generally associated with mathematical maturity.
I advise folks interested in economics to pursue an undergraduate degree in mathematics. If that is not feasible or realistic, then I suggest at a minimum some proofs-based work in linear algebra and real analysis. Friedberg's Linear Algebra text is a good exposition and will help you build up some maturity. It's a solid prerequisite for analysis. Once you start talking about the derivative as a linear operator, linear algebra is a must-have. However, linear algebra doesn't otherwise appear in real analysis before that.
For analysis, Abbott is friendlier than Rudin, and will likely provide you with sufficient background to move forward. Abbott is geared towards juniors in an undergraduate curriculum, while Rudin is the hard-core senior level text. If you don't have mathematical maturity, Rudin will force you to struggle. Once you have mathematical maturity, the conciseness his writing is quite nice.
Fudenberg and Tirole is a very technical, terse book. It is a good reference, but not my favorite exposition into game theory. I might be inclined to look through Maschler, Solan, and Zamir instead. It's gotten good reviews, though I haven't had a chance to look through it myself.So take this with a grain of salt. It is definitely more comprehensive than Fudenberg and Tirole, though.
I suppose I will suggest starting with Theory of Games and Economic Behavior by Neumman.
If I recall it, believe it or not, assumes very little mathematical background from the reader. Maybe you should check it out and see if reading it increases your proficiency with other works.
The reason that I think it will is that I'm not sure if the problem you're having is one of mathematics competency or one of comprehension. If it is the latter then you should know that authors tend to skip multiple steps in problem solving because they assume that you can figure things out on your own.
If this is where you're struggling, then the seminal book I mentioned is a good choice; because the authors are more forgiving and offer many examples. Looking back on it it has a whole section dedicated to a review of very basic set theory.
I think good preparation could be to watch some analysis video lectures. Examples this I think are quite good include
These lectures have the advantage that you get to (repeatedly) watch someone doing proofs and therefore learn about the kind of axiomatic reasoning that goes into putting together the kind of results you see in your graduate econ books.