Tom had the brilliant idea of not biting off too much at once with real analysis and studying pure proof techniques. He also found a very good book for it, it is even free. http://www.people.vcu.edu/~rhammack/BookOfProof/index.html.
We like it because it is geared at exactly our goal and requires little to no prerequisites.
We have been working on it in the past months quite slowly because of some personal things that needed attention, but now I am back and we are at Chapter 5.
It has not been a very difficult book for now, but it has been perfect at showing us the edge of how we currently think and how to extend it, formalized proof 'techniques' (which does not replace the need for or ), along with confidence in formal proof writing.
The book is not to be confused with the famous 'Proofs from the Book' ! we definitely do not have what it takes to tackle that one yet, but if at some point we do, we will see it as success of our project.
More good news: We have somebody (a colleague Technical Artist) who decided to join us and has been catching up starting at week 1. Unfortunately, we have not been commenting on his insightful posts, but that is the price of starting late :). We do have verbal discussions at work though...
I conclude with a Frankenstein of a proof I made, which has a much more elegant alternative found by Tom. It is a nice example of two different ways of proving the same thing, how the use of properties is often crucial to write elegant proofs, how I am using my 'debugging the truth' technique, how to be patient, how not assume that if you cannot see the path right away you will not be able to find it at all, and how you can learn from failed proofs instead of throwing the towel.