# How Important is Real Analysis

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My school offers three undergraduate courses (which are substitutes for one another) in real analysis, and I was wondering if taking one of the more difficult courses in real analysis would significantly help me getting into a top statistics graduate program?

The three courses offered are:

Math 444 - Elementary Real Analysis

Description: Careful treatment of the theoretical aspects of the calculus of functions of a real variable; topics include the real number system, limits, continuity, derivatives, and the Riemann integral. 4 hours of credit requires approval of the instructor and department with completion of additional work of substance. Credit is not given for both MATH 444 and MATH 447.

Math 447 - Real Variables

Description: Careful development of elementary real analysis including such topics as completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration. Credit is not given for both MATH 444 and MATH 447.

Math 424 - Honors Real Analysis

Description: A rigorous treatment of basic real analysis via metric spaces. Metric space topics include continuity, compactness, completeness, connectedness and uniform convergence. Analysis topics include the theory of differentiation, Riemann-Darboux integration, sequences and series of functions, and interchange of limiting operations. As part of the honors sequence, this course will be rigorous and abstract.

The elementary real analysis course is offered over the summer, which helps a lot since I'm completing all my my classes for a math major in 2 years (changed major), but I could take one of the other courses if it would really help my grad school chances. I'm also an honors student that meets the requirements for taking the honors class. Alternatively, I could take the elementary class this summer and the honors class in the fall.

Thanks

Edit: Math 444 uses Bartle, Math 447 uses Ross, Math 424 uses Rosenlicht

Edited by continued
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My school offers three undergraduate courses (which are substitutes for one another) in real analysis, and I was wondering if taking one of the more difficult courses in real analysis would significantly help me getting into a top statistics graduate program?

The three courses offered are:

Math 444 - Elementary Real Analysis

Description: Careful treatment of the theoretical aspects of the calculus of functions of a real variable; topics include the real number system, limits, continuity, derivatives, and the Riemann integral. 4 hours of credit requires approval of the instructor and department with completion of additional work of substance. Credit is not given for both MATH 444 and MATH 447.

Math 447 - Real Variables

Description: Careful development of elementary real analysis including such topics as completeness property of the real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration. Credit is not given for both MATH 444 and MATH 447.

Math 424 - Honors Real Analysis

Description: A rigorous treatment of basic real analysis via metric spaces. Metric space topics include continuity, compactness, completeness, connectedness and uniform convergence. Analysis topics include the theory of differentiation, Riemann-Darboux integration, sequences and series of functions, and interchange of limiting operations. As part of the honors sequence, this course will be rigorous and abstract.

The elementary real analysis course is offered over the summer, which helps a lot since I'm completing all my my classes for a math major in 2 years (changed major), but I could take one of the other courses if it would really help my grad school chances. I'm also an honors student that meets the requirements for taking the honors class. Alternatively, I could take the elementary class this summer and the honors class in the fall.

Thanks

Edit: Math 444 uses Bartle, Math 447 uses Ross, Math 424 uses Rosenlicht

Take the hardest one. Its a top course and you can never know it well enough. Also, some stats programs require a math sequence i.e. you might have to take it again anyway.

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If you want to study for a graduate degree in a top statistics program, you are going to eventually need to take a measure-theoretic course in probability theory. Of course, before you can take this class, you will need to have completed a year-long (usually) sequence in graduate analysis covering measure theory. Before taking this sequence in measure theory, you generally need at least a semester of rigorous undergraduate real analysis. Therefore, yes, you should take the hardest one.

Edited by etale
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Thanks guys.

Does the description of Math 424 look significantly better than the 447?

I'm now considering taking 444 (elementary) this summer and 424 (honors) in fall. I got killed when we did analysis last semester in my "fundamental mathematics" (intro to upper level maths/proofs) class.

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Real Analysis is extremely important. I got rejected from a program only because I didn't take the course. I then spoke to a committee member who said he would be willing to change the denied status to deferred it I promised to take the class.

So its very important.

I don't think it matters the level of the course, as long as you have it in your transcript, your'e good.

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If I take the "easy" analysis class this Summer, and then "hard" analysis class in the Fall (I only get to receive credit for one of the classes), does this look good or bad to an adcom?

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If you do well in both of them it will look good, if you screw up in the earlier one and do fine in the later one it will look ok, if you screw up in the later one, not so good

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Not my field, but I'd think that really knowing the material would be just as important as doing well. There have been classes that I've managed to do well in, but afterwards I couldn't really hold onto the material as well as I would have liked to because the material was just too hard (and too new and unfamiliar) for me to get really comfortable with in the time we had to work with it. So, if you think the material is going to be challenging, I think it makes A LOT of sense to take the elementary class first and the more advanced class after you've had an introduction to the concepts. You'd improve your chance of doing well in the more advanced class, and you'd certainly learn the material better, in my opinion, given that you'll have more time to work with it and think about it.

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Sounds good. I signed up for the elementary class in the Summer and the honors class in the Fall. Thanks.

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I'm at the same school as you and I would suggest you take 447 (it's not terribly difficult, PM me if you wanna which prof to shoot for) and supplement with the Rosenlicht book they've been using for the honors course. The honors version isn't incredibly difficult either, but there are some pretty time-consuming homeworks.

Either way, after taking one of these two, you will be able to take grad analysis over the summer as it's offered every summer and the person who usually teaches it is the best at teaching analysis at our school, surely you won't have the benefit of full semester's immersion in grad analysis, but this will prepare you to tackle any analysis you're going to need in grad school, I'd think.

And regarding the fundamental mathematics course you took where you had trouble, I had a similar experience. Don't let the fairly bad treatment of introductory analysis topics that they give in math 347 scare you away from the subject, you'll develop things with a much longer stride and with much more care in 447

Edited by barber5
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Thanks, sent you a PM.

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