I've personally read An Introduction to Mathematical Reasoning: Numbers, Sets and Functions Paperback by Peter Eccles and Chapter Zero by Carol Schumacher and would recommend them both.
Sadly, this is not an easy thing to learn and requires a lot of work and most importantly discipline. You mustn't let yourself be at all complacent. It's very easy when doing exercises by yourself to believe that you've "got the idea" and you can "see how it works", but the key is to actually write _everything_ down, so that there is no room for handwaving at all. This is not easy to force yourself to do, but it is necessary.
