Doing a PhD Economics without background in formal theorems? I could really use some advice. I am a new PhD student in Economics and Finance. My background is in economics, and while I did mathematics in my Master's course, it was more data-analysis based rather than formal proofs. So, for instance, I have become quite adept at analysing data in SPSS, STATA, R etc, but struggle when it comes to being given a formal problem and then having to prove using formal means; e.g. proving that a particular sequence is Cauchy or Martingale, etc.

What are the best ways to become better at formal theorem-proofs? I know that practice is the most obvious answer, but is there a base from which I should be starting? e.g. are there certain "bread-and-butter" proofs in mathematical economics such as law of large numbers, central limit theorem etc, that being able to prove would make it easier to prove others?

Would appreciate some advice on how to get better at the theorem-proof method, or even some textbooks/resources that would help with this, given that my background in this area is very limited.
I did my undergraduate double majoring in discrete mathematics and economics. I went through the grad micro sequence, the math for economists grad course, and the game theory field course, though. So I have a good idea of what the courses are like on both ends. (Doing the grad school route now, but that's additional commentary with which I won't bore everyone).

Traditionally, coursework like abstract Linear Algebra and Number Theory have been used by math departments to teach proof writing. In the last 15-20 years, an Intro to Proofs class has been added. Number theory doesn't come up appreciably in economics. So I'd punt that. But a solid course in linear algebra is so important. And if you have a strong data science background, then you should be very comfortable with the material, though perhaps not the formalization. 

Real analysis is the maturity sequence in the math departments. Rudin is the standard textbook. If you take it, expect to be putting in a solid 20 hours per week, but you will be razor sharp by the end of it. Real analysis is usually a senior or introductory grad sequence. There is often times an introductory analysis class that uses a text like Abbott. This introductory class is still something you shouldn't take lightly, but it should be much easier and sufficient for a lot of what you'll do in economics. If you are working in stochastic processes, martingales, etc., then measure is likely to come up, and so you should take the Rudin sequence at some point and follow it up with measure theory (which is a very tough grad course).

The math for economists course compresses much of these courses (usually not measure) into a semester. When I took it, we trucked through the abstract linear algebra semester-long course in three weeks. It was also an ugly presentation, but that's what happens when you push for speed rather than elegance. And this is one more reason why you should take these courses in the math department.

Combinatorics and graph theory is also becoming increasingly important in economics, with regards to networks. So a course in graph theory wouldn't hurt either. Graph theory proofs are picky in the sense that you have to be very precise in saying "because it looks like it." This is different than analysis and algebra proofs. Graph theory is my particular area, so I can definitely speak to it.

A course in analysis of algorithms or theory of computation wouldn't be a bad thing to have either, with complexity becoming increasingly important in economics (algorithmic mechanism design, algorithmic game theory, strategic network formation, showing computing equilibria is computationally difficult for certain classes of games, etc.). 

To sum it up, I'd strongly recommend:  
Abstract Linear Algebra  
Intro to Real Analysis

I'd recommend you consider: 
Real Analysis (Rudin) 
Measure Theory

Nice to Have courses: 
Graph Theory 
Combinatorics 
Theory of Computation 
Analysis of Algorithms

Let me know if you want book recommendations for any of these courses.