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[polymath257]

Fair enough. But know that I am always willing to answer questions. If you don't want full solutions, I can usually point a direction.

Welcome. There are a number of sock puppet creators that wander these here parts such that there is a certain cynicism about new members (I got a bit of that too), don't much worry about that. Proof of the pudding shall be in the eating thereof. I joined here on foot of chasing one such lying sock puppet around the internet (PGJ anyone?). PGJ, is a prolific internet crank. I have lost track of how many socks he has created here and on other sites. 

A close friend of mine is a PhD in math also. The stereotype annoys him as much as I suspect it does you.  Take it in good nature. My friends nickname is "traitor" because, while he worked at various interesting things such as weather/climate modeling, he now works for a seriously important financial institution. But no worries, another friend has a PhD in Biotechnology. His thesis was in the action of bacteria in the production of cheese of all things. Inevitably, he is known as "Dr. Cheese" for the last several decades.

It doesn't bother me excessively. It just sucks that a few idiots ruin things for everyone else. I was a bit surprised at the third degree I initially got from here, but I understand the reasons now.

[Abaddon_ire]

Welcome. There are a number of sock puppet creators that wander these here parts such that there is a certain cynicism about new members (I got a bit of that too), don't much worry about that. Proof of the pudding shall be in the eating thereof. I joined here on foot of chasing one such lying sock puppet around the internet (PGJ anyone?). PGJ, is a prolific internet crank. I have lost track of how many socks he has created here and on other sites. 

A close friend of mine is a PhD in math also. The stereotype annoys him as much as I suspect it does you.  Take it in good nature. My friends nickname is "traitor" because, while he worked at various interesting things such as weather/climate modeling, he now works for a seriously important financial institution. But no worries, another friend has a PhD in Biotechnology. His thesis was in the action of bacteria in the production of cheese of all things. Inevitably, he is known as "Dr. Cheese" for the last several decades.

It doesn't bother me excessively. It just sucks that a few idiots ruin things for everyone else. I was a bit surprised at the third degree I initially got from here, but I understand the reasons now.

Ah, you underestimate the power of the crank. Often, people here and elsewhere find themselves thinking "Is this yet another iteration of the same twit?" That happens. How could it not, given those willing to create any amount of sock-puppet accounts?

You are the captain of your soul. Just be yourself and be honest. Most here will accept that and give you honest argument in response. If you are a sock of some former member, then you will be found out in short order. If you are not, simply post whatever it is you wish to post and worry not a whit about it. Start a new thread about whatever it is that bakes your noodle about theism/atheism. Or whatever topic takes your fancy. See what happens.

You are learning the ropes for this site and we are learning your ropes as well. It is the tapestry of human interaction like it or not.

Gird your loins for robust responses from both sides. Such is the cut and thrust of online debate.

[polymath257]

It doesn't bother me excessively. It just sucks that a few idiots ruin things for everyone else. I was a bit surprised at the third degree I initially got from here, but I understand the reasons now.

Ah, you underestimate the power of the crank. Often, people here and elsewhere find themselves thinking "Is this yet another iteration of the same twit?" That happens. How could it not, given those willing to create any amount of sock-puppet accounts?

You are the captain of your soul. Just be yourself and be honest. Most here will accept that and give you honest argument in response. If you are a sock of some former member, then you will be found out in short order. If you are not, simply post whatever it is you wish to post and worry not a whit about it. Start a new thread about whatever it is that bakes your noodle about theism/atheism. Or whatever topic takes your fancy. See what happens.

You are learning the ropes for this site and we are learning your ropes as well. It is the tapestry of human interaction like it or not.

Gird your loins for robust responses from both sides. Such is the cut and thrust of online debate.

Oh, I have frequented a number of discussion forums in my time (I go back to Usenet). I am quite aware of the impact trolls, sock-puppets, and other such pricks.

And I enjoy robust responses. I just have to learn the local social customs.

[Kernel Sohcahtoa]

My apologies, polymath.  I'm afraid that I have zero ability/talent for math (nor am I any type of intellectual): I'm just a random guy who finds math interesting and wants to gain a basic understanding of it.  Based on your credentials, I'm pretty sure that I'd bore you or any serious student of mathematics.  With that said, there are certainly highly intelligent forum members here and members who are good at math, so I hope that you are able to meet them.  Thanks for joining AF, sir.

Fair enough. But know that I am always willing to answer questions. If you don't want full solutions, I can usually point a direction.

Recently, I've become interested in real analysis and would like to learn material that would typically be covered in an intro to real analysis course (e.g., sequences, series, continuity, differentiability, integrability, and the real numbers).  However, it seems that there's a lot of information to cover. With that said, in your opinion, what core concepts should I ensure that I understand in order to successfully complete the equivalent of an intro to real analysis course?  Also, what books or resources would you recommend? Thanks.

P.S. Would you happen to have access to a course syllabus that provides a clear trajectory of how to proceed through the course along with the key topics and concepts to study (perhaps books to get and core exercises to work in those books)? Thanks again.

[polymath257]

Fair enough. But know that I am always willing to answer questions. If you don't want full solutions, I can usually point a direction.

Recently, I've become interested in real analysis and would like to learn material that would typically be covered in an intro to real analysis course (e.g., sequences, series, continuity, differentiability, integrability, and the real numbers).  However, it seems that there's a lot of information to cover. With that said, in your opinion, what core concepts should I ensure that I understand in order to successfully complete the equivalent of an intro to real analysis course?  Also, what books or resources would you recommend? Thanks.

P.S. Would you happen to have access to a course syllabus that provides a clear trajectory of how to proceed through the course along with the key topics and concepts to study (perhaps books to get and core exercises to work in those books)? Thanks again.

Off the top of my head: be conversant with 'epsilonics': doing epsilon-delta or epsilon-N proofs. Be able to use the definitions of limits and continuity to prove basic facts (sum of a limit is the limit of the sum, the product of two functions continuous at a point is continuous at that point, etc). Know about suprema/infima of subsets of R.

Know various aspects of compactness (in the real line, this is the same as being closed and bounded): Heine-Borel, convergence of Cauchy sequences, every bounded sequence has a convergent subsequence (but be aware that these do not carry over to general metric spaces). Know about uniform continuity and how it differs from continuity. Know, however, that a continuous function on a closed interval is automatically uniformly continuous.

Know and be able to prove results like the Intermediate Value Theorem and the Mean Value Theorem. Be able to use the MVT to derive standard Calc I results about increasing/decreasing functions. Be able to prove that a differentiable function on an interval such that f'(x)=0 everywhere must be a constant function.

Extra Credit: know that it is possible to be continuous everywhere, but differentiable nowhere. Be able to give explicit examples for lack of differentiability at a single point.

Know the basic procedure for defining the Riemann integral. Be able to show that any continuous function is integrable. Know how to prove the Fundamental Theorem of Calculus for continuous functions.

Often this first course in real analysis is used as an 'intro to proof' class as well. So a lot of time is spent learning how to negate statements with quantifiers (there exists, for every) and others (negate an implication, for example).

As for books. The old standard was Rudin's 'An Introduction to Real Analysis'. If you can find Stromberg's book 'An Introduction to Classical Real Analysis', it is possibly just a bit higher level than what you want, but it is an excellent source of exercises. It doesn't do the Riemann integral, however, but goes directly into the Lebesgue integral. I'll ponder a bit for others that might be good and I'll see if I can find a course outline for you.

[Kernel Sohcahtoa]

Recently, I've become interested in real analysis and would like to learn material that would typically be covered in an intro to real analysis course (e.g., sequences, series, continuity, differentiability, integrability, and the real numbers).  However, it seems that there's a lot of information to cover. With that said, in your opinion, what core concepts should I ensure that I understand in order to successfully complete the equivalent of an intro to real analysis course?  Also, what books or resources would you recommend? Thanks.

P.S. Would you happen to have access to a course syllabus that provides a clear trajectory of how to proceed through the course along with the key topics and concepts to study (perhaps books to get and core exercises to work in those books)? Thanks again.

[1] Off the top of my head: be conversant with 'epsilonics': doing epsilon-delta or epsilon-N proofs. Be able to use the definitions of limits and continuity to prove basic facts (sum of a limit is the limit of the sum, the product of two functions continuous at a point is continuous at that point, etc). Know about suprema/infima of subsets of R.

[2] Know various aspects of compactness (in the real line, this is the same as being closed and bounded): Heine-Borel, convergence of Cauchy sequences, every bounded sequence has a convergent subsequence (but be aware that these do not carry over to general metric spaces). Know about uniform continuity and how it differs from continuity. Know, however, that a continuous function on a closed interval is automatically uniformly continuous.

[3] Know and be able to prove results like the Intermediate Value Theorem and the Mean Value Theorem. Be able to use the MVT to derive standard Calc I results about increasing/decreasing functions. Be able to prove that a differentiable function on an interval such that f'(x)=0 everywhere must be a constant function.

[4] Extra Credit: know that it is possible to be continuous everywhere, but differentiable nowhere. Be able to give explicit examples for lack of differentiability at a single point.

[5] Know the basic procedure for defining the Riemann integral. Be able to show that any continuous function is integrable. Know how to prove the Fundamental Theorem of Calculus for continuous functions.

[6]Often this first course in real analysis is used as an 'intro to proof' class as well. So a lot of time is spent learning how to negate statements with quantifiers (there exists, for every) and others (negate an implication, for example).

As for books. The old standard was Rudin's 'An Introduction to Real Analysis'. If you can find Stromberg's book 'An Introduction to Classical Real Analysis', it is possibly just a bit higher level than what you want, but it is an excellent source of exercises. It doesn't do the Riemann integral, however, but goes directly into the Lebesgue integral. I'll ponder a bit for others that might be good and I'll see if I can find a course outline for you.

Thanks for your thoughtful reply, polymath.   I've actually spent a great deal of time honing my understanding of proofs: I have worked numerous exercises in Solow's How to Read and Do Proofs, Hammack's Book of Proof, and Chartrand's Mathematical Proofs, all of which cover the basic proof writing techniques.  Regarding part [6] of your post, I gained a good understanding of learning how to prove statements  containing quantifiers (the choose, construction, and specialization methods) and how to negate statements containing quantifiers (and implications), especially nested quantifiers, via Solow, which IMO, is very useful when doing a proof by contradiction, contrapositive, or elimination.  

Regarding part [1] of your post, I gained a good understanding of epsilon-delta proofs and epsilon-N proofs via Chartrand's section on calculus proofs and was able to understand the proofs of the basic limit properties: I really think it is cool yet challenging to construct that respective delta or N value (after arbitrarily choosing an epsilon value greater than zero) that will ultimately result in the function or sequence being within epsilon of the limit. In addition, via Chartrand,  I did a number of exercises pertaining to properties of real numbers, especially the triangle inequality.  Also, via Abbott's Understanding Analysis (I own this), I have gained an understanding of suprema and infima of subsets of R and was able to wrap my head around the nested interval property, Archimedean property, and axiom of completeness. 

It sounds like most of my time will be spent learning parts [2], [3], [4] and [5] of your post (since I learned calculus, I always wanted to learn why integrals work).  Also, I do own Rudin's Principles of Analysis; it seems quite advanced, but nevertheless, it does look like it is an invaluable resource
. I also own Introduction to Real Analysis by Schramm (I like his proof writing style, as it resembles Solow); Introduction to Analysis by Rosenlicht; Foundations of Analysis by Johnsonbaugh, and How to Think About Analysis by Alcock.  With that said, thanks for taking the time to respond to me. 

P.S. Since my goal is to learn material covered in an intro to real analysis course, would you recommend that I also study metric spaces or is this topic more advanced than the material that is usually covered in an intro level course? Thanks again for your time and attention.

[Abaddon_ire]

Ah, you underestimate the power of the crank. Often, people here and elsewhere find themselves thinking "Is this yet another iteration of the same twit?" That happens. How could it not, given those willing to create any amount of sock-puppet accounts?

You are the captain of your soul. Just be yourself and be honest. Most here will accept that and give you honest argument in response. If you are a sock of some former member, then you will be found out in short order. If you are not, simply post whatever it is you wish to post and worry not a whit about it. Start a new thread about whatever it is that bakes your noodle about theism/atheism. Or whatever topic takes your fancy. See what happens.

You are learning the ropes for this site and we are learning your ropes as well. It is the tapestry of human interaction like it or not.

Gird your loins for robust responses from both sides. Such is the cut and thrust of online debate.

Oh, I have frequented a number of discussion forums in my time (I go back to Usenet). I am quite aware of the impact trolls, sock-puppets, and other such pricks.

And I enjoy robust responses. I just have to learn the local social customs.

So far, to me you seem genuine. But that is provisional. In other posts, it appears to me you are an archetype of seeking a PhD paper.

There is nothing per se wrong with that. But if that is your goal, honesty would yield a more productive paper.

[polymath257]

P.S. Since my goal is to learn material covered in an intro to real analysis course, would you recommend that I also study metric spaces or is this topic more advanced than the material that is usually covered in an intro level course? Thanks again for your time and attention.

In my opinion, metric spaces are better studied as a precursor to topology. But this depends on how it is done and what the later goals are. Other mathematicians disagree and have good reasons for doing so.

It isn't so much that the material is more advanced as it is more abstract. Many of the proofs from real analysis carry over *almost* verbatim to metric spaces. And while others fail, that failure can clarify what is going on in the real analysis. The nice thing is that doing metric spaces opens up the analysis of function spaces and puts uniform convergence of functions in its 'proper' setting.

On the other hand, questions of differentiability and integrability simply don't arise in metric spaces. The focus is on issues related to continuity between very general spaces. So if your goal is to understand derivatives and integrals, it is probably best to wait for metric spaces until after real analysis. If, instead, you want to understand questions concerning continuity, uniform continuity, Cauchy sequences, compactness, and completeness into proper perspective, do some metric spaces. Metric spaces also open up the study of Banach spaces and Hilbert spaces, which are large subjects in themselves.

---

Oh, I have frequented a number of discussion forums in my time (I go back to Usenet). I am quite aware of the impact trolls, sock-puppets, and other such pricks.

And I enjoy robust responses. I just have to learn the local social customs.

So far, to me you seem genuine. But that is provisional. In other posts, it appears to me you are an archetype of seeking a PhD paper.

There is nothing per se wrong with that. But if that is your goal, honesty would yield a more productive paper.

If I stumble across a good *physics* PhD topic, and I could get one of the local physics profs to be my advisor, I wouldn't complain. But I don't expect that to happen.

Do you really have a problem with PhD candidates wanting to write about this forum?

[Abaddon_ire]

P.S. Since my goal is to learn material covered in an intro to real analysis course, would you recommend that I also study metric spaces or is this topic more advanced than the material that is usually covered in an intro level course? Thanks again for your time and attention.

In my opinion, metric spaces are better studied as a precursor to topology. But this depends on how it is done and what the later goals are. Other mathematicians disagree and have good reasons for doing so.

It isn't so much that the material is more advanced as it is more abstract. Many of the proofs from real analysis carry over *almost* verbatim to metric spaces. And while others fail, that failure can clarify what is going on in the real analysis. The nice thing is that doing metric spaces opens up the analysis of function spaces and puts uniform convergence of functions in its 'proper' setting.

On the other hand, questions of differentiability and integrability simply don't arise in metric spaces. The focus is  on issues related to continuity between very general spaces. So if your goal is to understand derivatives and integrals, it is probably best to wait for metric spaces until after real analysis. If, instead, you want to understand questions concerning continuity, uniform continuity, Cauchy sequences, compactness, and completeness into proper perspective, do some metric spaces. Metric spaces also open up the study of Banach spaces and Hilbert spaces, which are large subjects in themselves.

---

So far, to me you seem genuine. But that is provisional. In other posts, it appears to me you are an archetype of seeking a PhD paper.

There is nothing per se wrong with that. But if that is your goal, honesty would yield a more productive paper.

If I stumble across a good *physics* PhD topic, and I could get one of the local physics profs to be my advisor, I wouldn't complain. But I don't expect that to happen.

Do you really have a problem with PhD candidates wanting to write about this forum?

Not in the slightest.

[polymath257]

Just curious: what is the etiquette concerning reputation points? I some forums, it is 'polite' to thank the person giving such in PM. Is that the case here?