How Can I Ease Into Understanding Mathematical Proofs?

[Nano-Passion]

I was very discouraged when I couldn't do a couple proofs myself in calculus such as the squeeze theorem. My textbook has very little steps into some of the proofs and assumes that the student should infer most of the information.

Not being able to follow the proofs made me feel that I hated them. But I went to khan academy and followed their proofs and it was much more helpful! I liked the squeeze theorem proof immediately.

I need a source that would help ease my way into proofs and help me mature mathematically. I'm very new to the whole proof concept. Any ideas?

 

[micromass]

It seems to me that you don't have any problems with proofs, but that you simply have trouble following the $\epsilon-\delta$ context?? Am I right??

I remember when I did my first $\epsilon-\delta$ proofs, it was horrible. I understood everything just fine, but the epsilon-delta stuff just didn't work. It took a long time before I finally made sense of it.

I suggest you make a lot of exercises on epsilon-delta proofs. In the beginning, it's rather awkward because you use inequalities that you've never seen before, but in the end it's really easy.

Take a decent calculus book like Spivak and do some epsilon-delta things. And watch Khan academy quite a lot.

Are there other proofy things that are bothering you??

 

[leright]

when I first saw the squeeze theorem I didn't think it was possible to prove it. It just seems like common sense. Then the prof proved it and I thought that a proof just wasn't necessary. But that's what a lot of rigorous math is...proving things that are common sense.

 

[Nano-Passion]

It seems to me that you don't have any problems with proofs, but that you simply have trouble following the $\epsilon-\delta$ context?? Am I right??

I remember when I did my first $\epsilon-\delta$ proofs, it was horrible. I understood everything just fine, but the epsilon-delta stuff just didn't work. It took a long time before I finally made sense of it.

I suggest you make a lot of exercises on epsilon-delta proofs. In the beginning, it's rather awkward because you use inequalities that you've never seen before, but in the end it's really easy.

Take a decent calculus book like Spivak and do some epsilon-delta things. And watch Khan academy quite a lot.

Are there other proofy things that are bothering you??

My apology, I was aiming my topic about proofs and inadvertently put $\epsilon-\delta$ definition. I understand its context now, and it was rathar awkward as you say in the beginning. What I really meant to talk about was the proof aspect of things like

$lim_{x--> 0}\stackrel {sin x}{x} = 1<br />$

Not terribly good at latex right now but you get the point (sinx/x)

I don't know what proofs I'm having trouble with exactly because I haven't done much proofs at all. I'm just having a hard time with proofs in general. If you give me a random conjecture to prove right now, I will most likely not be able to do it without help.

I want to be at a level where I can look at the proof problems at the end of the section and not be completely lost on what to do.

 

[QuarkCharmer]

Those sin/cos limits have non epsilon delta proofs using l'hospital's rule. I still have little to no idea what an epsilon delta is! Anyway, they are right, Spivak's calculus is really proofy. I'm taking the 3-course calculus set now using Stewart and I like to check out how Stewart explains things and look at the practice problems in that book. I also purchased "How to prove it" at the recommendation of some fellow pf posters and it's pretty great, but it went over my head really quickly. I think the book is more tuned for people who about proofs, but need to improve. I have yet to find a decent "introduction" book..

 

[micromass]

Well, you could buy Velleman's "how to prove it", but that won't help you with epsilon-delta. It's made for another kind of proo.

To be honest, if you would ask me how to prove

$$\lim_{x\rightarrow 0}{\frac{\sin(x)}{x}}=1$$

(press quote to see what I did for LaTeX!), then I wouldn't know how to start either! Of course, such a limit is easy to prove through l'hospital's rule or Taylor series. But I wouldn't know how to start a good epsilon-delta proof. And the thing is that you're not expected now to prove such a things point-blank.

The things you should be able to prove now are things of the sort

$$\lim_{x\rightarrow 1}{x^2}=1,~\lim_{x\rightarrow 1}{\frac{1}{x}}=1,~\lim_{x\rightarrow a}{g(x)f(x)}=g(a)f(a)$$

These things are not difficult but require practice. To be able to be a good prover, you must first

1) Read a lot of proofs
2) Understand those proofs
3) Try some very easy cases
4) Do some harder stuff

Limits like $\frac{\sin(x)}{x}$ are already quite hard, so nobody expects you to be able to do this.

Get Spivak and read some epsilon-delta arguments and try some easy modificiations of the arguments. That's the only way to learn!

 

[micromass]

Those sin/cos limits have non epsilon delta proofs using l'hospital's rule. I still have little to no idea what an epsilon delta is! Anyway, they are right, Spivak's calculus is really proofy. I'm taking the 3-course calculus set now using Stewart and I like to check out how Stewart explains things and look at the practice problems in that book. I also purchased "How to prove it" at the recommendation of some fellow pf posters and it's pretty great, but it went over my head really quickly. I think the book is more tuned for people who about proofs, but need to improve. I have yet to find a decent "introduction" book..

Quark, what about this book: www.maths.manchester.ac.uk/~nige/IMRpartI.pdf
It's free and it looks quite easy!

The thing also is that you can't separate proofs from their context. A proof book will be quite hard until you see the natural context of the proof. Sure, you can prove things like "if n is even, then so will n²", but that's boring.
Ideally, one would learn proofs while learning another part of math.

 

[Nano-Passion]

Well, you could buy Velleman's "how to prove it", but that won't help you with epsilon-delta. It's made for another kind of proo.

To be honest, if you would ask me how to prove

$$\lim_{x\rightarrow 0}{\frac{\sin(x)}{x}}=1$$

(press quote to see what I did for LaTeX!), then I wouldn't know how to start either! Of course, such a limit is easy to prove through l'hospital's rule or Taylor series. But I wouldn't know how to start a good epsilon-delta proof. And the thing is that you're not expected now to prove such a things point-blank.

The things you should be able to prove now are things of the sort

$$\lim_{x\rightarrow 1}{x^2}=1,~\lim_{x\rightarrow 1}{\frac{1}{x}}=1,~\lim_{x\rightarrow a}{g(x)f(x)}=g(a)f(a)$$

These things are not difficult but require practice. To be able to be a good prover, you must first

1) Read a lot of proofs
2) Understand those proofs
3) Try some very easy cases
4) Do some harder stuff

Limits like $\frac{\sin(x)}{x}$ are already quite hard, so nobody expects you to be able to do this.

Get Spivak and read some epsilon-delta arguments and try some easy modificiations of the arguments. That's the only way to learn!

Thanks, that was uplifting. Do you have something a bit cheaper? Spivak is a bit on the expensive side as of now. I'll go to my library and see if they have it though.

 

[BruceW]

I want to be at a level where I can look at the proof problems at the end of the section and not be completely lost on what to do.

As you said in your first post, you enjoyed the squeeze theorem proof when you saw it on kahn academy. So maybe try a few proofs from there, or from other books. When you do more, you'll get better at doing them.
If your textbook assumes the student should infer the information, then maybe its because being able to do the proofs are not 100% necessary for your course. But being able to do them is a good thing anyway. Or maybe they are necessary. Maybe you could ask your lecturer if you'll get examined on them.

For the sinx/x proof: If you've done taylor series, then you should be able to do it. (Or l'hospital's rule). I guess the tricky bit is thinking "what maths do I need to use to prove this?" And then making the connection.

 

[micromass]

Thanks, that was uplifting. Do you have something a bit cheaper? Spivak is a bit on the expensive side as of now. I'll go to my library and see if they have it though.

Check out http://hbpms.blogspot.com/2008/05/stage-2-calculus.html for many good introductory real analysis books. Most of these books contain some stuff on epsilon delta.

 

[Nano-Passion]

As you said in your first post, you enjoyed the squeeze theorem proof when you saw it on kahn academy. So maybe try a few proofs from there, or from other books. When you do more, you'll get better at doing them.
If your textbook assumes the student should infer the information, then maybe its because being able to do the proofs are not 100% necessary for your course. But being able to do them is a good thing anyway. Or maybe they are necessary. Maybe you could ask your lecturer if you'll get examined on them.

For the sinx/x proof: If you've done taylor series, then you should be able to do it. (Or l'hospital's rule). I guess the tricky bit is thinking "what maths do I need to use to prove this?" And then making the connection.

Proofs aren't required one bit and won't be on the exam. Sometimes I find that proofs help give me that intuition behind a concept. Furthermore, it always gives me an appreciation for the mathematics. Its really easy to study if you are interested in something. Its better to ease my way into proofs now then to go crazy later in a proof-heavy course.

I agree, the hardest part is knowing how to start.

 

[Nano-Passion]

Check out http://hbpms.blogspot.com/2008/05/stage-2-calculus.html for many good introductory real analysis books. Most of these books contain some stuff on epsilon delta.

Why an introductory to real analysis books? I don't really know what real analysis is about but I would guess that calculus is a prerequisite. I'm only up to calculus I.

 

[micromass]

Why an introductory to real analysis books? I don't really know what real analysis is about but I would guess that calculus is a prerequisite. I'm only up to calculus I.

Because most calc I books do not bother with epsilon-delta stuff (except for books like Spivak and Apostol, but you found them too expensive). So the material for epsilon-delta proofs is often contained in the real analysis books. That's why I gave those.

Of course, calculus is a prereq for real analysis books. But if you only look up specific things like epsilon-delta things, then this won't hurt you.

 

[Nano-Passion]

Because most calc I books do not bother with epsilon-delta stuff (except for books like Spivak and Apostol, but you found them too expensive). So the material for epsilon-delta proofs is often contained in the real analysis books. That's why I gave those.

Of course, calculus is a prereq for real analysis books. But if you only look up specific things like epsilon-delta things, then this won't hurt you.

Thanks, I'll look at the epsilon-delta, but I'm more worried about learning proofs in general.

 

[micromass]

Thanks, I'll look at the epsilon-delta, but I'm more worried about learning proofs in general.

Check out my link in post 7, that should contain a nice introduction to proofs! But it won't help you with epsilon-delta stuff...

 

[flyingpig]

Honestly I still have trouble writing my inductive hypothesis...

 

[QuarkCharmer]

Honestly I still have trouble writing my inductive hypothesis...

This is the type of thing that I am looking for help in.

 

[Nano-Passion]

Check out my link in post 7, that should contain a nice introduction to proofs! But it won't help you with epsilon-delta stuff...

Okay.

Honestly I still have trouble writing my inductive hypothesis...

Never heard of the term before . Is it the same as this? http://en.wikipedia.org/wiki/Mathematical_induction

 

[Cuauhtemoc]

Not very advanced, or advanced at all, but the old "what is mathematics?" book by R. Courant explains very well the idea of mathematical proofs and some basic examples.
What is mathematical induction, proof by contradiction, direct proof...
It may be of some help.
He also got a very good book on calculus, and a lot easier(in my humble opinion) than spivak's.

As I went to eng school I didn't really learn much about proving things, I knew how to compute integrals and use matrix's but never understood WHY. This book opened my eyes a lot, even though it could be read by a high-schooler (a motivated one).