How to get hang of mathematical language that is so extensively used in economics? 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.