Proof writing is an essential skill for graduate econ, especially the micro theory sections. Proof writing is one of the key skills you learn from a real analysis course, but there is also specific content that is important. For example, proofs of continuity, closure, boundedness are all standard micro-theory fare. You will use open balls, Cauchy sequences, and other concepts from analysis in your first year micro classes, and it's a lot easier to learn the economics if you don't have to struggle through the underlying math at the same time.
You might want to take a look at MWG if you haven't already. If you have any familiarity with analysis, you'll recognize that a lot of the essential concepts are employed in the basic econ.
Bottom line is that there is a reason that current econ students and professors advise you to take analysis. They really do know something you don't.