Need some advice regarding apostol calculus volume 1

[glb_lub]

Hello, I am new to doing mathematics of the type in Apostol. I wish to study analysis. Hence I have started with Apostol Calculus volume 1 so as to build my fundamentals.
I need a tip in solving the exercise problems which one encounters in first few chapters.

I am presently on the introductory chapter.

My query is, while proving a theorem such as :- 'there exists a real number z such that x < z < y for any arbitrary x , y' , is it ok to prove that z = (x + y)/2 meets the condition. Apostol never really makes use of '2' in his proofs upto this point. For this I will have to first prove that 2 > 1 , I think. Or should I stick with arbitrary real numbers such as 'a' .

This issue is also relevant while proving that there is an irrational between any two rationals.
Can I make use of √2 in such cases ? I will prove √2 is irrational and then proceed to make use of it and its reciprocal.

For a beginner level such as mine , are such proofs considered valid ? And at what level should one strive for elegance ?

I had browsed some analysis books in the past and I found they never make use of particular numbers such as '2' and '√2' and rather make use of general properties of the numbers involved.

 

[chiro]

Hey glb_lub and welcome to the forums.

The nature of the proofs in analysis type subjects are going to be things like proving convergence or proving things with norms, metrics and the like.

Like you were saying in your post, we will assume that we have structures of some sort (represented by some kind of set) where we will assume that the structure has certain conditions that have to be followed.

For example inner products will have specific conditions as will norms and metrics. Convergence has a specific definition that has to be followed and can be written in terms of norms. Different kinds of norms can be related through inequalities (like Minkowski).

You will be focused more on these kinds of things rather than proving say things about SQRT(2) being irrational.

In saying this, working with sequences and series is very important in analysis and you will also look at the consequences of working with infinite series, infinite sized 'vectors' and the equivalent of an infinite sized operator (matrix).

When you look at this, it will get complicated and you will have to think a little differently than if you were looking at finite sized spaces because everything like convergence requires a lot more careful analysis and more constraints.

It might help if you post some of the stuff you will be working on to get more specific advice.

 

[glb_lub]

It might help if you post some of the stuff you will be working on to get more specific advice.

Well , I have really only just begun , so can't really say what I will be working on. I am currently reading introductory texts such as Apostol - Calculus, Volume 1.

Currently I am doing the exercises in the Introductory chapter.

I am facing the a problem of the following nature :-

In Apostol , the positive integers are introduced as those real numbers which are members of every inductive set.
In the exercises that follow , one is asked to 'prove that an integer is either odd or even' . I could go some distance in solving it but in the end I had to make use of a fact that '1 is the smallest positive integer'. Now Apostol doesn't give us any major properties of the integers other than that they are members of every inductive set. I tried to use this fact to prove 1 is smallest positive integer but couldn't prove it.
In a later section , while proving the well ordering principle , Apostol mentions this fact that '1 is smallest positive integer but doesn't go to prove it (or at least I thought it wasn't proved).

In one or two earlier problems also I had to use this fact that '1 is the smallest positive integer'. Are my proofs legit ?

I also couldn't make use of the principle of mathematical induction , since it comes after two sections in this book.

So , in short many times I am not sure whether I can assume some mathematical theorem/fact and use it in my proof ,since it hasn't been stated as an axiom specifically or proved in a theorem. Like in the above example , I am unsure whether I could use the fact that '1 is smallest positive integer'.