Mathematical Induction / Proof writing / Real Analysis topics

[BerenTebogo]

Was going to post in the LHC thread but I know there are a lot more purely math guys in here than just those who post in LHC.

Long story short, I'm really struggling MASSIVELY for the first time since going back to college with the concepts in my Intro to Real Analysis I class. I guess it's primarily just a lack of coherently understanding the concepts, but as the class goes on the uncertainty about one topic snowballs into the next topic.

I've tried to get help from some professors but nobody is really saying anything that is giving me that "AH HA!" spark. In fact some of the professors I have talked to who specialize in PDE/ODE, higher calculus, etc, just say that they basically hated this shit and can't really help me, lol.

We're still on some pretty basic stuff (I think) but we're moving away from the 'intro' chapter into the meat of the course and if I don't get this shit now it's going to get bad real fast. Some concepts like Archimedean Property, which I get the basic form of n*epsilon > p, or whatever, I just don't see how it's so immensely useful in other proofs. Covered and uncovered sets, deleted neighborhood(?), open / closed subset and the like really just kinda...go right over my head. Hell, even just writing the proof itself kinda gets me.

I'm hoping that some of you here maybe had similar experiences and found a way to get past them, or a way of understanding these ideas that I might be able to benefit from. Or hell, maybe some of you would just be better at explaining than my book / professor (very likely.)

Any help, no matter how small, is appreciated.

[Woozie]

I was a TA for Advanced Calc/Intro to Analysis I and II last year and I'm TAing for them this year as well. I've seen many students who struggled with these classes, and these are the things that seemed to work best for them.

I've noticed that the students who search for solutions to homework assignments online seemed to do well on tests, even though that's technically cheating. They typically used those as a way to understand the solutions though, so it's almost as if they had a bunch of extra worked examples.

Don't be afraid to go to your professor or TA for help a LOT. Go as much as you need to and get every single homework problem checked and explained to you if you can. Work with another student or two as well.

There are youtube lectures for these courses that can be very useful. Sometimes they'll explain things in a way that's different from your professor and that could be just what you need. For actual Real Analysis (i.e. not the undergrad intro version) I had to use youtube to understand compact sets because the book didn't explain it very well and I pretty much never went to class.

I can't believe some of the people you went to in this area hated this stuff. My top 4 favorite classes of all time are Real Analysis, Advanced Partial Differential Equations II, Advanced Calculus/Intro to Analysis I, and part II. I chose to write my thesis on partial differential equations/hyperbolic conservation laws because this area of math is just so much fun to me.

I'm still here to help, as usual. I'm not on facebook as often as I used to be and I'm a LOT busier than usual this semester, but I'll still do what I can. If you have skype or something, it may be easier for me to explain stuff to you that way. If you ever need help, feel free to message me on facebook (not on here, because I never notice PMs) and I'll get back to you when I'm free. It may be a better idea to post a topic here or post in the LHC thread though since I may be too busy to help. There's quite a few other people here who have taken these classes as well and can help.

[Woozie]

And just out of curiosity, did you have some form or proof writing class before taking this class? Those make a HUGE difference. If you haven't, you may want to find a textbook used in such a course (such as "A Transition to Advanced Mathematics") as a supplement. I'm sure you can download the book online for free somewhere.

You may also consider downloading or going to the library for other books for your same subject. That's what helps me the most when I'm learning physics. I almost always have two or three books on the same subject.

[Ferion]

I really struggled with Advanced Calc when I took it. What I found really helped was what Woozie mentioned about online solutions. I wouldn't necessarily look up homework questions, but there are a ton of sample questions online that have very clear explanations behind them along with the reasoning used for each step. The biggest thing is to not get frustrated and take your time going over the material, the class you're taking is one of the most difficult undergraduate math classes there is in my opinion. Proofs really aren't my strongest area in math, but I'll try to answer any question you have the best I can.

[Kaylia]

Real analysis is probably one of the hardest classes, but also one of the most important, If you're struggling, don't panic and keep doing your best, you will eventually understand it and it's going to help you considerably.

One thing I noticed about Analysis and proof writing classes is that most people who do well at the exam don't understand what they are doing at all. They are just mimicking everything they saw without understanding the content. This is a major difference with physics here you need to understand the concept to applies them. It also means you have to approach studying differently if you want to succeed.

If you're worried about failing a test, try to memorize the complete proof (or method) even if you don't understand it at all. Of course, do this as a last recourse, but keep that in mind if you need to buy some time and grde.



Concerning the content, it's hard to give you an universal advice, but if you have a particular question (anything you're struggling with), just ask, we will be more than happy to help you with it.



[edit]
Where exactly do you have trouble? Do you have trouble with the language itself (how to read a proof/mathematical sentence)? Do you have trouble with the concept mentioned? If you do, just wiki the one you come across. they are all very simple, and often relate to stuff you have done in high school.


If you have trouble with the proof itself, don't worry, they are all very touchy, and it's nearly impossible to do all of them on your own without knowing the "trick".

[BerenTebogo]

And just out of curiosity, did you have some form or proof writing class before taking this class? Those make a HUGE difference. If you haven't, you may want to find a textbook used in such a course (such as "A Transition to Advanced Mathematics") as a supplement. I'm sure you can download the book online for free somewhere.

You may also consider downloading or going to the library for other books for your same subject. That's what helps me the most when I'm learning physics. I almost always have two or three books on the same subject.

I have ZERO background in proof writing but I've been picking it up as I go along. Still trying to get used to the fact that every little thing that I normally take for granted needs to be said explicitly, but yeah.

I think looking at solutions would be helpful, the problem is that we're not really given 'homework problems' to do. My professor makes us look at theorems and proofs in our book with their definitions and then construct full arguments for them (proofy type) and it's just hard as all fuck for me.

Real analysis is probably one of the hardest classes, but also one of the most important, If you're struggling, don't panic and keep doing your best, you will eventually understand it and it's going to help you considerably.

One thing I noticed about Analysis and proof writing classes is that most people who do well at the exam don't understand what they are doing at all. They are just mimicking everything they saw without understanding the content. This is a major difference with physics here you need to understand the concept to applies them. It also means you have to approach studying differently if you want to succeed.

If you're worried about failing a test, try to memorize the complete proof (or method) even if you don't understand it at all. Of course, do this as a last recourse, but keep that in mind if you need to buy some time and grde.



Concerning the content, it's hard to give you an universal advice, but if you have a particular question (anything you're struggling with), just ask, we will be more than happy to help you with it.



[edit]
Where exactly do you have trouble? Do you have trouble with the language itself (how to read a proof/mathematical sentence)? Do you have trouble with the concept mentioned? If you do, just wiki the one you come across. they are all very simple, and often relate to stuff you have done in high school.


If you have trouble with the proof itself, don't worry, they are all very touchy, and it's nearly impossible to do all of them on your own without knowing the "trick".

I understand the language fairly well. This semester was the first time I've ever seen any of it really but it's pretty straightforward. Lots of the concepts, specifically neighborhoods, closed and open sets, deleted neighborhoods, coverings, etc just kinda go over my head with what the book is telling me. Some of the basic reasoning behind proofs by contradictions kinda make my head spin sometimes too if it's covered too quickly.

I'm sure that I can do well on the test from just strict memorization but that's not all I want out of the class. I am just getting really discouraged by stuff like the definition of limit points and stuff, a concept I thought I understood inside and out, screwing with my head.

I think I'll look for the book on basic intro to proof writing as a resource, but do you guys have any good websites with worked out solutions to these things that I could use as a guide? The wiki stuff doesn't help me since I'm comfortable with the notations used in class, that level of strict mathematical notation just kinda makes my eyes gloss over (on wiki).

Edit: Kaylia, got a bigger version of your avatar? I want, lol. Just finished steins gate, I loved it.

[BerenTebogo]

Also woozie, I kinda love DE now. My PDE class is very challenging and I'm struggling to keep up (3 person class, the guy flies through, and nobody really understands what he's doing but because of my experience in quantum I think I have a leg up on all this fourier nonsense). I think I'm gonna take numerical PDEs next semester if I can A this course.

[Kaylia]

I understand the language fairly well. This semester was the first time I've ever seen any of it really but it's pretty straightforward. Lots of the concepts, specifically neighborhoods, closed and open sets, deleted neighborhoods, coverings, etc just kinda go over my head with what the book is telling me. Some of the basic reasoning behind proofs by contradictions kinda make my head spin sometimes too if it's covered too quickly.

Edit: Kaylia, got a bigger version of your avatar? I want, lol. Just finished steins gate, I loved it.

Avatar or sig? Well, here is both
http://imageshack.us/photo/my-images...ps1qz83i8.jpg/
http://imageshack.us/photo/my-images...paper2842.jpg/


Concerning the concepts you are having issue with, you're probably overthinking and look for explanations that are not needed. They are all extremely simple, and the information you need is always included in the definition.

In analysis, all you have are "axioms", "definitions" and "theorems". The axioms will be listed at the very beginning of the book and generally involve basic operation (ie: addition, multiplication, 0, 1, cardinality). This is all the knowledge you need to prove every single theorems that follow.

Definitions (or concepts) are used to simplify the writing, but there is nothing more to them other than the literal meaning. If they are talking about "open set", it was most likely defined previously, and that definition is -everything- you need to know (nothing more, nothing less). Don't try to find another meaning, and don't think of open set as a deep concept, simply replace the word "open set" with its definition every time you come across it. You still need to understand the definition in relation to previously established concept, but you don't need to understand how this definition will be used in the future.

And finally, there is theorems that are rules that are proven using axioms or already proven theorems (but ultimately, they all go back to axiom). They can be hard to prove, but these classes are generally made in such a way that you will know how to do it easily.


I know it sound stupid when I put it like this, but understanding the axiomatic nature of mathematics is 90% of the work. Once you grasp that structure properly, everything will become much clearer.

I'm sure that I can do well on the test from just strict memorization but that's not all I want out of the class. I am just getting really discouraged by stuff like the definition of limit points and stuff, a concept I thought I understood inside and out, screwing with my head.

One advice I could give you is to not worry too much about linking the concept you see in that class to concept you have learned before. In the end, they are the same thing, but what you're trying to do here is a bottom-up approach of mathematics, and for that, you don't need concept establish in other class, just what is established since the axioms.




[edit]

Also, there really isn't many different types of proofs (4-5? I forgot). It's very important to understand them, and it can be confusing the first time, but to be honest, once only have to "get them" once, because it's the same thing every time after.

Contradiction is by far the worse in my opinion (it's like reading a sentence with double negative) , but if you think about it a little, it will make sense.

[BerenTebogo]

Yeah the double negative business is a mind fuck right now, idk. I just got back from a study session and I was able to work through some of the stuff I needed for this assignment, which is good. I think you've got a point Kaylia that I'm trying too hard to find some deep, substantial meaning to all of these things when some of them really are just literally what they sound like.

I'm having a lot of problems though with trying to use stuff like the archimedean property in proves to formally PROVE what I'm saying. Or really any of the theorems, I keep just wanting to write out a sentence and be like SEE IT MAKES SENSE!

[Kaylia]

I had the same issue when I was writing proof. Hell, most physicists seem to struggle since we are "programmed" to use our intuition and common sense to solve problems, and it's not something we can do here.



Archimedean properties is usually written like this
If x>0, y is real, a "n" exist so that n*x > y
and it will always be true for real number

If at some point, this inequality is proven wrong (no "n" can possibly satisfy this equation), the statement you're trying to prove will always be wrong (and if its a proof by contradiction, it means the original statement is always right)

Sometime, what you want instead is the value of "n" when n*x finally surpass y. Then you input it somewhere else to confirm the statement is respected,


Without a particular example, it's hard to tell you how it's used, but the trick is to read every line one by one and try to understand where they are come from, and if they make sense. Once every single line make sense, you can go back and understand the proof as a whole.