### Undergraduates in real analysis and graduates who have taken real analysis: when did it finally click for you?

get abbott's understanding analysis if this is not already what you're using. (there are also solutions available in different places...) sometimes seeing a different presentation of real analysis makes it "click" more readily. abbott's book is much gentler than some of the other texts out there.

work with other people. if their pace is faster than yours, then work on your own for "quite a few hours" beforehand. one part of analysis that i remember as being a bit tough was the rapidity with which new ideas seemingly emerged: epsilons & deltas, continuity, closed and open sets, cluster points [or accumulation points or limit points], sequences, series, convergence & divergence & cauchy, uniform continuity, least upper bounds & greatest lower bounds, intermediate value theorem, etc.

and often ideas have been cleaned up [over the last ~150 years] and -- on purpose! -- they are presented in a reversed order. this can make "proofs" look pretty opaque. [incidentally, if you ever take a course on real analysis part ii: i recommend the text by H.S. Bear for its readability.]

i recommend trying to prove something in multiple ways when possible. e.g. if you have multiple definitions of compact then try proving the same statement with each definition. or take a proof of the form "if blah then BLAH" and prove its contrapositive.

e.g. "if C is closed, then it contains all of its limit points."

well, prove the contrapositive (what is the contrapositive?!):

"if C does not contain all of its limit points, then it is not closed."

[note that the opposite of closed is "not closed" ... as opposed to the frequent confusion of sets being either closed or open; but they could be neither, and the interval [0,1) readily serves as such an example.]

i won't write out a proof of the statement or its contrapositive above, as i don't know if you've arrived there by week 6 -- and because there are multiple ways to define closed & to define limit point. [in fact, the statement is really iff; so some people define a closed set in R as one that contains all of its limit points!]

& when you can (perhaps with help from a classmate or an instructor or TA or posting to math.stackexchange): see if you can find in a proof what is the concept being used, and what are the details. not an easy dichotomy to identify as a beginner.

anyway. note that i am suggesting something a bit strange: you are having trouble with a class, and rather than suggesting you spend more time on the material covered in the class, i am actually suggesting that you spend more time looking at the related material outside of the class. finding a real analysis text that isn't your own. solving problems in more than the way done in your book/class. if you can, try to re-prove the theorems you read: definitely in your own words, but perhaps even using your own method (e.g. by switching to a different definition).

& go to office hours.