From Engineering-Style to Proper Mathematics I currently have an engineering-style education in mathematics. We covered quite a lot of material (e.g. real and complex analysis, some probability theory and graph theory), but more often than not we stated theorems without formal proofs, and, what's worse, there was a significant amount of hand waving (moving limits around without proper justification, and the like).
Now I am coming to the realization that this approach is akin to going on steroids to build your muscles: while it lets you cover a lot of ground in a relatively short time, it creates more problems than it solves in the long term. 
What I would like are suggestions to 'convert' my mathematics knowledge into proper knowledge, using a self-study approach complemented with questions on here. What I have in mind are questions like (but not limited to):


*

*Where should I start? (set theory?, predicate logic?).  

*What follows? (real analysis? complex analysis?). 

*Should I work through whole books, solving every problem? 

*What are the core topics that any self-respecting mathematician should know well?

 A: Two possibilities:


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*Take what you have learned for granted (yes, there are holes; but what you got explained by vigorous handwaving can be proved rigurously). Dig down to the formal proofs of new material as you need it, perhaps filling in whatever underlying material makes you uneasy.

*Start from the beginning, fill in any and all holes.


As a matter of expediency, I'd go with (1), (2) will take you a long time to get to where you want to go. In fact, I have found (by pesonal experience) that trying to learn preemptively tends to lead you astray by not having a goal in mind. Besides, it often happens than when you need something already studied, you'll have forgotten; or, perhaps even worse, you don't ever need it.
A: To understand the basics of most of the math used in engineering, I suggest you start with real analysis, then move on to complex analysis, functional analysis and measure theory. Real analysis might be a bit hard at first, if you're not used to finding and writing rigorous proofs, but it'll get easier afterwards.
