One way to proof an algorithm is, find something that is always true when the program iterates over. You need to: 1.prove that the condition is true before start; 2.proof iterations will not change it.

And it sounds very similar to Proof by Induction in math. Maybe some math proving guide is helpful too.

I think a rigorous introductory class on algorithms would be good. You should be able to find such MOOC courses on Coursera, MIT OpenCourseWare, ...

This is a really great text on the said topic :-

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2005/lecture-notes/l2_proofs.pdf