Is this definition of infinite limits needlessly restrictive?

Question

I am studying James Stewart's "Calculus: Early Transcendentals 7th Edition". On page 115, a precise definition of infinite limits is presented:

Let $f$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then

$\lim_{x \to a}f(x) = \infty$

means that for every positive number $M$ there is a positive number $\delta$ such that

if $\hspace{1em}0 < |x - a| < \delta \hspace{1em}$ then $\hspace{1em} f(x) > M$

I am trying to understand why Stewart is limiting $M$ to the set of positive numbers. This choice feels arbitrary and superfluous. It seems to me that Stewart could just as easily have said "for all $M > 42$".

Is the word "positive" in the phrase "for every positive number $M$" required in this definition? Why not just say "for every number $M$"? Would the definition be any less precise?

Answer 1

Any one of these phrases renders the definition unchanged:

1. "for every $M\in \mathbb{R}$"
2. "for every positive number $M$" (original definition)
3. "for every $M>42$" (undoubtedly, you chose this number as it is the answer to the Ultimate Question of Life).

It should be clear the definition with 1. implies 2. implies 3.

The definition with 3 implies 1 because if there is a $\delta$ such that $|x-a|<\delta\implies f(x)>42$, then the same $\delta$ works for any $M\leq 42.$ So they are all equivalent.