# Thread: Real Analysis: what the hell is it used for in Economics?

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## Real Analysis: what the hell is it used for in Economics?

Many Econ grad students and professors I have talked to, upon mentioning I want to go to grad school in Econ and questioning what math I should be prepared for, have always highly recommended taking Real Analysis. But the thing is that I have read up a little on it, and it seems to mainly contain elements of set theory and proofs of formulas I learned over two semesters of Calculus.

Now set theory and Calculus, I know why they are useful in Economics (well except for all the trigonometry). So does not just taking undergrad Discrete Structures and 3 semesters of Calculus deliver much more utility on the margin than taking a class which proves all of that material? If developing a mind for proofs is the objective, is it not more useful just to take a course in proof structures (which I have been told is not that useful for graduate Econ in any case)?

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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.

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Well, of course, everyone will also tell you that you need to take three semesters of calculus, linear algebra, and a decent prob/stats course, too.

But as for real analysis, you are correct about the material. You should be familiar, on a hand-waving level, with everything it covers: limits, continuity, set theory, etc. But the unique thing, what sets it apart from the calculus classes you've taken, is that being familiar with the concepts on an intuitive level is not enough. You have to be able to prove, rigorously, everything you already know. This comes more easily to some than others, but for everyone, it should be a very, very different course than, say, differential equations. Any reasonably competent person with enough time on her hands can memorize what rule to apply in what situation, well enough to take care of differential equations on the level for that class. Fewer people are able to apply the knowledge they have in new ways to prove concepts of calculus. Demonstrating that you have this ability signals a lot of things that an A in differential equations (or, for that matter, a course called something like "proof structures," which is usually taught on a much lower level than real analysis) couldn't signal.

This is important because PhD-level econ is taught in a very different way from undergrad econ. No long is the emphasis on being able to draw some graphs and describe which way some policy or action might move the lines. You need to be able to prove the convexity of a curve. Prove that it's strictly decreasing over a certain range of factors. And unlike in undergrad courses, you have to do all this without numbers. You won't have numbers to plug into your calculator, so you'll be forced instead to prove everything. Trying to do this without having taken real analysis is not something I would recommend.

I would really urge you to track down a copy of MWG and look through it some to get a feel for the proposition-proof-interpretation layout of the book. It feels very different from an advanced undergraduate textbook, because the approach is very different.

4. Good post? |
Many Econ grad students and professors I have talked to, upon mentioning I want to go to grad school in Econ and questioning what math I should be prepared for, have always highly recommended taking Real Analysis. But the thing is that I have read up a little on it, and it seems to mainly contain elements of set theory and proofs of formulas I learned over two semesters of Calculus.

Now set theory and Calculus, I know why they are useful in Economics (well except for all the trigonometry). So does not just taking undergrad Discrete Structures and 3 semesters of Calculus deliver much more utility on the margin than taking a class which proves all of that material? If developing a mind for proofs is the objective, is it not more useful just to take a course in proof structures (which I have been told is not that useful for graduate Econ in any case)?
I always felt the same about trigonometry while learning calculus. But I have to admit that some functions may be helpful when studying economic cycles

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On the original topic, I am reminded of a quote from my multivar calc prof: "Numbers are useless after primary school." (imagine at 9am w/a heavy Russian accent, and it becomes funny). I am working through Rudin independently this summer (advice on this process is welcome), and I can definitely see how the proof techniques are applicable to micro and econometrics.

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Originally Posted by wolf87
On the original topic, I am reminded of a quote from my multivar calc prof: "Numbers are useless after primary school." (imagine at 9am w/a heavy Russian accent, and it becomes funny). I am working through Rudin independently this summer (advice on this process is welcome), and I can definitely see how the proof techniques are applicable to micro and econometrics.
wolf87, is this your first introduction to formal proofs? If so, you might consider a book called "How to Prove It" by Velleman. For more general mathematical reasoning, try "The Art and Craft of Problem Solving." I don't remember the author but can look it up if you can't find it.

I think it's a lot easier to internalize the material in Rudin if you simultaneously study its applications. My advice would be to work through a section of Rudin, and then read and try to recreate the proofs in MWG that use the analysis material you've just learned. If recreating the proofs is too difficult (and don't worry, they are often quite complicated) then instead, try to convince yourself of where the proof would fail without the crucial assumption or step (convexity, continuity, boundedness, etc.) Hope this helps.

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Thnx asquare. I have seen some formal proofs before, but a review would be very useful.