Is real analysis really that hard?

[SMA_01]

I'm a sophomore math major, and I' currently taking proofs, linear algebra (not proofs-based), and calc 3. These classes aren't that bad so far. I met with a math adviser today, and he told me for my major requirements I should take real analysis 1&2, Linear algebra, and abstract algebra for a strong math background. I'm worried I am not cut out for these classes though. Are they that hard? I feel like I'm not "mature" enough in math to handle these courses :/

 

[deluks917]

Thy are not unreasonably hard.

 

[micromass]

They are not hard at all. You just need to work a little bit harder for them than you did for calc. All those classes are proof based, so you'll want to know a bit of proofs. Not much, because you'll learn things along the way.

Take the classes and be sure to solve a lot of exercises and ask a lot of questions (even if you think you know the answer!). This forum could help you with it.

 

[mathwonk]

in suspect one reason analysis is considered so hard is that for a long time there has been a tradition of teaching it from rudin and ahlfors, very user unfriendly books. find another book you like and see what i mean. the ideas are hard enough, but the presentation makes a big difference. try anything by berberian or lorch or wendell fleming.

 

[eliya]

I'm taking Real Analysis right now, and I think it's certainly do-able. I think it boils down to whether you like spending a lot of time reading, re-reading, trying to understand what you read, then do some problems, and then do even more. I think that this is what it essentially boils down to. The concepts are hard and challenging, but that's what's making this class so much fun.
Other than that, I think that the level of difficulty can change from one professor to another. Some can assign original or just harder problems, and some assign easier problems. Many of the problems in our textbook are just known theorems that you need to prove. That makes them a little easier, because I can always read about the solution somewhere else (though I try not to). But the ideas behind analysis that is taught in university x by professor x_1, is the same as the ideas taught in university y by professor y_1.

Granted, I haven't finished my Real Analysis class yet, but I'm doing very well. Here are my tips for you:
Spend a lot of time with the material, and spend even more time solving problems.
Do all the assigned problems, and try to do the problems that aren't assigned too. Sometimes you get assigned problems 4, 6, 9, and 12, but the ideas proved in 1-3,5,7-8 can help you with the assigned problems.
Have more than one textbook. The textbook they use in your class is probably fine, but it's nice to have other sources to learn from. Sometimes a certain idea or theorem is proved better in one book than the other.
Lastly, practice on how to write proofs. My intro to proofs class was ok. It taught me the ideas behind some proofs techniques, but the problems were very straightforward and kind of computational. You'll need more than just that for Real Analysis. Make sure you really know how to write proofs, because I think that's a main thing that prevents many students in my class from succeeding. They are capable of understanding the material, but they kind of suck at writing proofs, but only due to a lack of practice!
Good luck!

 

[Poopsilon]

Rudin is a right of passage, don't accept anything less.

 

[eliya]

try anything by berberian or lorch or wendell fleming.

https://www.amazon.com/gp/product/0387984801/?tag=pfamazon01-20
Seems like it is a graduate textbook that isn't suited for undergrads.

 

[mathwonk]

using rudin is sort of like being paddled to join a fraternity, not really essential or recommended.

 

[mathwonk]

if berberian is too hard, try this one:

Introduction to Topology and Modern Analysis (International Series in Pure and Applied Mathematics) Simmons, George F.

but never dismiss a book because of the stated audience - look at it to see if that audience includes you.

 

[mathwonk]

try this by berberian:

https://www.amazon.com/dp/0387942173/?tag=pfamazon01-20

 

[mathwonk]

making something look hard that should be easy is not really the sign of a great teacher.

 

[daveb]

I personally liked https://www.amazon.com/dp/0471159964/?tag=pfamazon01-20 Fundamental Ideas of Analysis when I learned it.

 

[mathwonk]

the calculus books of spivak and apostol are excellent places to learn rigorous ideas that are often considered part of elementary real analysis.

 

[lurflurf]

Like many subjects real analysis can be as easy or hard as you make it. A few things to keep in mind are real analysis is concerned with functions that are yucky to varying degrees, try not to assume functions are always nice. Real analysis is about using a few central ideas, try to see this.