How can we compare a number with inifinity?

Question

we know that $a<\infty\space\space \forall a\in\mathbb{R}$ but why? how can order in $\mathbb{R}$ include this concept?

Answer 1

Often, when we use the $\infty$ sign, it is convenient shorthand for something else.

In calculus, we might write $\lim_{x\rightarrow a} f(x) = \infty$, as shorthand for saying that the limit fails to exist, and in particular that it fails to exist in such a way that as $x\rightarrow a$, the values of $f$ become larger than any finite amount.

The ancient Greeks used to speak this way about infinity. Instead of saying "there are infinitely many primes", they would characteristically say something more like "The collection of all primes is greater than any finite collection you could provide."

Nowadays, we might compare different infinities and include them in our sets, such as $\overline{\mathbb{R}}$, the extended real line. In some settings, it's convenient to have $\infty$ as a genuine manipulable object in your set. In other settings, it's convenient to use $\infty$ as a shorthand symbol when saying that a function, a sequence, etc. grows larger than any finite bound.

Answer 2

If $<$ is used to denote the usual ordering on $\mathbb{R}$, then $a < \infty$ is indeed nonsensical.

But we can consider orderings on larger sets, such as the usual ordering on $\overline{\mathbb{R}}$.

We use the same symbol for both the usual ordering on $\mathbb{R}$ and the usual ordering on $\overline{\mathbb{R}}$ since nearly every situation falls into one of two categories:

• Which ordering is intended is clear from context.
• It doesn't matter which ordering is intended. (e.g. when asking if $7$ is less than $5$, both orderings say "no")