The meaning of the symbol $\infty$ in Spivak's calculus book Spivak in "Calculus" writes 


  ... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$.


What is the meaning of being purely suggestive? 

Source:
$\infty$ and $-\infty$ are not real numbers. So when ever we use them in real analysis we need to define their use carefully, as he has done in the extract you quote.

For instance the interval $(a,b)$ is defined as $\{ x \in \mathbb R \ : \ a &lt; x &lt; b \}$ whenever $a, b$ are both real numbers. 

This definition is in trouble if the symbols $a$ or $b$ are $-\infty$ or $\infty$. In those cases, we need a different definition, as Spivak gives. This is to say, up to that point of Calculus, we know what $$x &lt; y$$ means for real numbers $x$ and $y$. We have not yet defined what for instance $-\infty &lt; x &lt; b$ would mean.

This might seem pedantic, especially in the example of interval notation, but mathematical results can often turn on carefully delineated definitions and expressions. So it's important to get even these basic definitions right in one's early training in rigorous mathematics, which is what Spivak's Calculus is all about.

In addition to the use of $\pm\infty$ in interval notation, Spivak will later give careful definitions  of expressions such as

$$\lim_{x\to\infty} f(x),$$ $$\lim_{n\to\infty} a_n,$$

$$ \lim_{x\to a} f(x) = \infty \quad\text{ where $a$ is a real number,}$$

and

$$\int_a^\infty f(x) \ dx \quad\text{ where $a$ is a real number}.$$

Those definitions will not treat $\infty$ as a real number. Instead, the definitions are expressed using real numbers in such a way that the suggestion of $\infty$ is formalized.
You may have heard the adage, "infinity is not a number, it's a concept" (or at least you have now). This is in fact true, since $\infty$ is not a real number (if it were, terrible things would happen). 

Infinity $-$ $\infty$ $-$ is the object such that no number is greater than it:
$$ \forall x\in \mathbb R : |\infty| &gt; |x|$$

We use this object to talk about the "extreme" ends of the real numbers (e.g., when talking about taking limits), because it is easier to say "as $x \to \infty$, foobar is true" than it is to say "for any [possibly very large] number $M \in \mathbb R$ there exists a number $N&gt;M$ such that foobar is still true".