How to succeed in Real Analysis?

[phillyolly]

I am taking Real Analysis I this semester. I am blown away with its difficulty. Proofs are so hard to comprehend, I am at total loss...

I am seeking for your advice on how to understand Real Analysis I for a beginner. What kind of learning techniques should I use to comprehend the material?

Thank you.

 

[nonequilibrium]

Always tie the abstract notions with examples (think of Q, R, the empty set -- simply sets that embody some special characteristic), in that way you will also get some intuitive sense of the notion. This will lead you in trying to make your own proofs.

In a sense, the abstract definition of something shouldn't be a word in your head that has a definition, but should be some sort of concept that embodies a certain intuitive understanding, as that makes it a lot easier to tie other notions to that concept and by time you will see a whole structure appear.

More practically: make a lot of exercises. The reason is the same as above. Do NOT neglect this! You might not feel any progress in the beginning, but the exercises help you get feeling for the notion, something no definition can ever do.

A helpful thing to remember: nothing useless is defined, so even though at the moment of seeing the definition you might feel totally unconnected to what it says, as if it's the most useless and far-away piece of text ever, do not think so.

If you have some more specific questions, I'd like to help.
For the rest: good luck! And enjoy :) I've had my first course in real analysis this past year and I thoroughly enjoyed it.

 

[HallsofIvy]

First and foremost- learn the definitions. And I don't mean "have a general idea what they mean", learn the specific words of the definitions. All definitions in mathematics are "working" definitions. You use the precise words in the definition of concepts in proof about those concepts.