I typically study on my own and while I find that I'm more likely to remember things if I write something down, there's an absurd amount of information in any halfway decent analysis text, even an intro text. This is the advantage of taking an actual course; someone with experience can wittle the info down to the essentials. So my approach is to do something else instead:

As I read through the text, I look for things that make me ask "why?". I'll sit back and think on this and then, either in the margin or on a piece of paper (detailing context in this case) I'll work out the answer. I've found that, for me at least, this works pretty well and is especially helpful in remembering and understanding proofs.

I'll also find examples for certain claims. For instance, if my text claims that for a bijection f:X->Y from an at-most countable set X, the image of X is at-most countable, I might find a specifc bijection that satisfies this claim.

If you want to write down what appear to you to be vital results, try writing them in your own words. Examine what you've written and see if you can find any errors in the way you've written these results and see if you can correct them.