I am reading the book "A Treatise on Advanced Calculus" by Philip Franklin. I found this book in our city's central library and liked it at the first reading, am continuing this book as a reference. I got some confusion at the very first topic "Mathematical Induction". I am familiar with mathematical induction pretty much but I could not grasp the way author explained it.
In the link i given $-$ After introduction the first topic starts. It is divided in $8$ small paragraphs. I understand $7$ paragraphs out of these. I have confusion in the $3$rd paragraph which I am quoting here:
"If each of the members of $I$, an infinite collection of positive integers, is less than, or equal to, $N$, every integer of the collection $I$ is equal to some member of the finite collection $1, 2, .\ .\ . N$ Thus there is a finite collection of distinct integers, $F$, such that each member of $I$ is equal to a member of $F$. The greatest integer of $F$ is the greatest of $I$, so that the first collection $I$ has a greatest integer."
The two points on which I have trouble are as follows:
- How this paragraph satiates(offsets) the whole article, that is what is the purpose of this paragraph. In other words what is the logic provided by this paragraph which is necessary for an apt understanding of the remaining article. What is the benefit of assuming that $I$ has a greatest integer.
- Why the author assumes that $I$, an infinite collection of positive integers, has an upperbound(a greatest number)? We know that $I$ is an infinite set so has not any greatest integer. Moreover how can we disprove this argument, that is does it directly follows from the definition of $I$ that it does not have a greatest integer or it can be proved in a mathematical way?
I understood the remaining $7$ paragraphs. I am not able to understand what the author is trying to convey in this $3$rd paragraph that I quoted.