Understanding Analysis

[Anonymous No.5466182]

So I'm studying Real number analysis from "Understanding analysis" by stephen abbott, and I don't understand it. I don't know how to "proof" something, can some one explain or send me to a source where I can start easy and build my self up.

The book is great, its just I don't understand proofs, it seems almost pointless to me, and I need to change my view on it, since it's getting in my way of finishing this course.

I do great in my physics and calc courses (85%-100% graduating grades) but I just don't understand. it's not a matter of "pleb, just doesn't understand math"... I just have to learn it, and I feel once I get over this first step I should be just fine.


like for proving that square root of 2 is not a natural number since if you consider prime numbers +1 ... for some reason that proves that it isn't a natural number.

I understand why square root of 2 isn't a natural number, I'm having trouble proving it in a way that seems logical to me.


Thanks for all the help

[Anonymous No.5466225]

I'll assume you're familiar with the Peano axioms.

Using these axioms, we can prove that the set {0, S(0), S(S(0)), ... } = {0, 1, 2, ... } is a subset of the set of natural numbers, where S(a) = a+1 is the successor function.
Using these axioms, we can prove, using induction, that the set of natural numbers is a subset of the above set. Thus the sets are equivalent.
We can also establish an order on N, and we can use the above fact to prove that, for any natural number n, there does not exist a natural number x such that n < x < S(n).
But if x^2=2 and x is a natural number, then 1 < x < 2. This is a contradiction, therefore x is not a natural number.

You will not need an equal level of rigor (from the Peanos) for the rest of analysis.

[Anonymous No.5466241]

>>5466182
Do you mean prove it's irrational? I've never seen someone trying to prove it's not natural.
Either way, you just need to get used to proves. Read them, read them again, and read them once again until you get the logical steps involved. Learning about logic is also useful.

[Anonymous No.5466252]

>>5466241
I plan to do that, thanks!
as for the logic of " oh hey! I'm going to use induction!" does that come with time or is there a pattern that I will find?

[Anonymous No.5466266]

>>5466252
Patterns that come with time

[Anonymous No.5466274]

is there such a thing as "easy" proofs to start with to start building intuition, or it doesn't matter.

can you guys post links to wikis or sites that discuss this type of stuff

thanks!

I'm heading to bed, so I'll respond in the morning.

[Anonymous No.5466308]

>>5466182
>So I'm studying Real number analysis from "Understanding analysis" by stephen abbott, and I don't understand it.

For some reason this made me lol.

OP, proofs are generally covered earlier on as well. The idea is that if you start out with only axioms you should be able to build the rest of mathematics on top of it with proofs showing the truth of other arguments and turning them into theorems.

In modern mathematics we hardly ever (pretty much never) talk about mathematical systems that aren't axiomatized. It's important to understand it as well, because sometimes (as you've surely seen with quantum mechanics) our intuition will betray us. Sometimes we'll even be dealing with mathematical systems completely different from the standard ones where you'll have to proof things that you have very little intuition for and in general difficulty having a hard grasp on. Proofing is so important to mathematics that many people will argue that anything that does not involve proofs is not mathematics.

Take this argument for example and instead of the proof (mathematical method) apply the scientific method. Every number is smaller than 1,000,000. Why is this not true? If you started to test numbers it would be true for the first 999,999 numbers you tested. Of course, in this case you can obviously see it's not true. How about something a little less obvious.

n^2 + n + 41 will always come out to a prime number if you insert an integer for n.

[Anonymous No.5466325]

>>5466308
Not OP but why not?
Let n = 41k for k integer. then n^2 + n + 41 will be divisible by 41 => not prime.

[Anonymous No.5466333]

>>5466274
Found this on google, probs sort of cheating
http://www.proofwiki.org/wiki/Main_Page
Click on random proof, look at general process, contribute if you feel so compelled (liberate information! etc).

>wait until tomorrow
inb4 404

[Anonymous No.5466356]

>>5466345
So?

[Anonymous No.5466435]

>>5466325

The point was that in the example.

>n^2 + n + 41 will always come out to a prime number if you insert an integer for n.

You can insert values for n one by one using the scientific method. It's kind of tedious and most people will give up sooner or later just assuming that it's true. This equation will output a prime number all the way until n = 41 (I think it may work for another number a little bit smaller but don't remember off the top of my head).

To see why this is so plug 41 in.

41^2 + 41 + 41
(41)(41) + (2)(41)
>consider that AC + BC = (A+B)C by the distributive property
(41+2)(41)
(43)(41)
>Which is clearly not prime

Your 41k example also works, but is a little bit less obvious.

There are many even less obvious scenarios in mathematics where people will convince themselves that something must be true when only really sophisticated proofs can show them otherwise. Sometimes this is referred to as mathematical coincidence.

[Anonymous No.5467049]

That is why most schools have a proofs course before algebra/analysis. Search UCSD Math 109, plenty of material to help you out.

Also Harvey Mudd has a complete real analysis 1 course online.