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I have of late had the chance to go through Chapter 1 of each of the following two books:

  1. Principles of Mathematical Analysis by Walter Rudin, 3rd edition

  2. Mathematical Analysis by Tom M. Apostol, 2nd edition

The first chapters of both of these books is about the real and complex number systems, the extended real and complex number systems, and $\mathbb{R}^n$.

In Definition 1.23, Rudin defined the extended real number system as the real field $\mathbb{R}$ and two symbols, $+\infty$ and $-\infty$, with the definitions that, for every $x \in \mathbb{R}$, $$-\infty < x < +\infty, \ \ \ x+\infty= +\infty, \ \ \ x-\infty = - \infty, \ \ \ {x \over +\infty} = {x \over -\infty} = 0, \ \ \ .$$ If $x \in \mathbb{R}$ and $x > 0$, then $$x \cdot (+\infty) = + \infty, \ \ \ \mbox{ and } \ \ \ x \cdot (-\infty) = - \infty.$$ On the other hand, if $x \in \mathbb{R}$ and $x < 0$, then $$x \cdot (+\infty) = - \infty, \ \ \ \mbox{ and } \ \ \ x \cdot (-\infty) = + \infty.$$

That's all that Rudin defines.

Now in Definition 1.24 in his book, Tom M. Apostol defines the extended real number system as the set of real numbers together with two symbols $+\infty$ and $-\infty$ which satisfy the following properties.

(a) If $x \in \mathbb{R}$, then we have $$x+(+\infty) = +\infty, \ \ x+(-\infty)= -\infty, \ \ x-(+\infty) = - \infty, \ \ x-(-\infty)=+\infty, \ \ {x \over +\infty}= {x \over -\infty}=0.$$ (b) If $x > 0$, then we have $$x(+\infty)=+\infty, \ \ \ x(-\infty)=-\infty.$$ (c) If $x < 0$, then we have $$x(+\infty)=-\infty, \ \ \ x(-\infty)=+\infty.$$ (d) $$(+\infty)+(+\infty)=(+\infty)(+\infty)=(-\infty)(-\infty)=+\infty.$$ $$(-\infty)+(-\infty)=(+\infty)(-\infty)=-\infty.$$ (e) If $x \in \mathbb{R}$, then we have $-\infty<x< +\infty$.

Although both the books are about mathematical analysis, Apostol assumes much more than Rudin does about $\pm\infty$. Why?

What is the justification for these discrepencies?

Which definition is the more standard one?

How does Rudin manage to do without what Apostol defines in part (d) above?

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These "definitions" are to be considered merely as conventions and should not be mixed up one with another but looked upon as abbreviations. $\infty$ and $-\infty$ are used as symbols and the given rules are shorthands for expressions representing certain limits.

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  • $\begingroup$ have you gone through this material in these particular books? And, as Rudin has done, can we dispense with what Apostol has defined in part (d) of his definition? $\endgroup$ – Saaqib Mahmood Dec 25 '15 at 14:09

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