Yes, you can try to prove everything before it's explained with pretty much any real analysis book. Just be reasonable about it, if you've gone a few hours without even making partial progress on a theorem, read the proof. A first exposure to analysis doesn't just teach you analysis, it teaches you how to build theories from the bottom up. If you can do that on your first try, great. If you can't (as is a lot more likely), learn how and save the "prove everything on your own" experience for a different subject.