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Understanding Real Analysis

Hi there, first year undegrad here. I am coping well with most things but it appears that real analysis i something that I do not seem to be understanding at all!

The recommended books don't really seem to be helping much or explaining in friendly terms and my professor does not break things down at all. The proofs seem to come out of nowhere for most part, and I do not understand some of the thought processes. The worst part seems to be knowing where to start and how. I have no idea what to do or how to get my feet on the ground right now.

Does anybody know any simple texts? Any that I've read have been assuming that I understand a lot in the first place.
(edited 9 years ago)
Original post by TheBBQ
Hi there, first year undegrad here. I am coping well with most things but it appears that real analysis i something that I do not seem to be understanding at all!

The recommended books don't really seem to be helping much or explaining in friendly terms and my professor does not break things down at all. The proofs seem to come out of nowhere for most part, and I do not understand some of the thought processes. The worst part seems to be knowing where to start and how. I have no idea what to do or how to get my feet on the ground right now.

Does anybody know any simple texts? Any that I've read have been assuming that I understand a lot in the first place.

Tim Gowers's texts are great. http://gowers.wordpress.com/category/cambridge-teaching/ia-analysis/
In particular, http://gowers.wordpress.com/2014/02/03/how-to-work-out-proofs-in-analysis-i/

I myself have written some things:
http://www.patrickstevens.co.uk/archives/474/
http://www.patrickstevens.co.uk/archives/584/

Hope they might help. Generally, it's just a matter of practice - write down what you know, and pattern-match.
Reply 2
Avatar for TheBBQ
TheBBQ
OP
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Original post by Smaug123
Tim Gowers's texts are great. http://gowers.wordpress.com/category/cambridge-teaching/ia-analysis/
In particular, http://gowers.wordpress.com/2014/02/03/how-to-work-out-proofs-in-analysis-i/

I myself have written some things:
http://www.patrickstevens.co.uk/archives/474/
http://www.patrickstevens.co.uk/archives/584/

Hope they might help. Generally, it's just a matter of practice - write down what you know, and pattern-match.


Thank you, I'll look into these tommorow as I've already had 6h of lectures today :tongue:

Do you have any tips for newbies when it comes to understanding and learning analysis? It's so completely different to everything I have done before.
Original post by TheBBQ
Thank you, I'll look into these tommorow as I've already had 6h of lectures today :tongue:

Do you have any tips for newbies when it comes to understanding and learning analysis? It's so completely different to everything I have done before.

Remember that beginner's analysis is so easy that even computers can do it. It's just that you have no experience in it that makes it seem hard. If you end up trying to do something difficult, you've probably chosen the wrong path.

Whenever you see a theorem, try and understand what the theorem means. Only secondary is what it actually says. Most theorems you'll meet will have only one core idea, and most of the steps surrounding that idea will be "forced": there is only one way to get from the starting information to that idea.

Do you have an example of a problem you've found difficult (preferably one you have finished, but it took you a long time to do)? I'd like to demonstrate.
Reply 4
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TheBBQ
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Original post by Smaug123
Remember that beginner's analysis is so easy that even computers can do it. It's just that you have no experience in it that makes it seem hard. If you end up trying to do something difficult, you've probably chosen the wrong path.

Whenever you see a theorem, try and understand what the theorem means. Only secondary is what it actually says. Most theorems you'll meet will have only one core idea, and most of the steps surrounding that idea will be "forced": there is only one way to get from the starting information to that idea.

Do you have an example of a problem you've found difficult (preferably one you have finished, but it took you a long time to do)? I'd like to demonstrate.


I'll try to keep that advice in mind.

First that seems to have lost me (and many people) is this guy's proof that the square root of 2 is irrational. I am not sure how finding a smaller integer would suddenly mean that the root of 2 is irrational? Reading it again and again it sort of makes a bit of sense but not quite.



There are other things which are not really explained, such as this:



I understand about the integers part, but why does the divisor have to be a natural number? Surely it can just be another integer?
Reply 5
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0x2a
2
Original post by TheBBQ
I'll try to keep that advice in mind.

First that seems to have lost me (and many people) is this guy's proof that the square root of 2 is irrational. I am not sure how finding a smaller integer would suddenly mean that the root of 2 is irrational? Reading it again and again it sort of makes a bit of sense but not quite.



There are other things which are not really explained, such as this:



I understand about the integers part, but why does the divisor have to be a natural number? Surely it can just be another integer?

For the proof of 2 \sqrt2 's irrationality, the author has claimed that q∈Z q \in \mathbb{Z} is the smallest positive integer so that 2q∈Z \sqrt2 q \in \mathbb{Z} . Now later on in the proof the author finds that q(2−1)<qq(\sqrt2 - 1) < q, and that 2q(2−1) \sqrt2q(\sqrt2 - 1) is also an integer, but we already defined that q q , which is larger than q(2−1)q(\sqrt2 - 1), is the smallest positive integer such that 2q∈Z\sqrt2q \in \mathbb{Z}, thus we have reached a contradiction and 2\sqrt2 cannot be rational.

Basically the author defined aa to be the smallest integer such that a2∈Za\sqrt2 \in \mathbb{Z}, but then he finds a b<a b < a , such that b2∈Z b\sqrt2 \in \mathbb{Z} .

For the notation it is simply because having the denominator be an integer allows the denominator to be 0 which obviously causes problems.
(edited 9 years ago)
Reply 6
Avatar for TheBBQ
TheBBQ
OP
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Original post by 0x2a
For the proof of 2 \sqrt2 's irrationality, the author has claimed that q∈Z q \in \mathbb{Z} is the smallest positive integer so that 2q∈Z \sqrt2 q \in \mathbb{Z} . Now later on in the proof the author finds that q(2−1)<qq(\sqrt2 - 1) < q, and that 2q(2−1) \sqrt2q(\sqrt2 - 1) is also an integer, but we already defined that q q , which is larger than q(2−1)q(\sqrt2 - 1), is the smallest positive integer such that 2q∈Z\sqrt2q \in \mathbb{Z}, thus we have reached a contradiction and 2\sqrt2 cannot be rational.

Basically the author defined aa to be the smallest integer such that a2∈Za\sqrt2 \in \mathbb{Z}, but then he finds a b<a b < a , such that b2∈Z b\sqrt2 \in \mathbb{Z} .

For the notation it is simply because having the denominator be an integer allows the denominator to be 0 which obviously causes problems.


That makes a lot more sense :smile: thank you!
Reply 7
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Asmeeta
Hey there! I did all that. I can help

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