Yes or No: The following $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ holds

Question

Can someone verify whether $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ is mathematically rigorous?

Answer 1

The infinity symbol $\infty$ has at least a dozen different meanings according to context. In each context, one must guess which kind of infinity is intended by the author, if the author has not explicitly stated the meaning. In this case, I guess that what you are talking about is the elements $\infty$ and $-\infty$ of the extended real number system, which is an algebraic/analytic structure based on the set $\bar R=R\cup\{\infty,-\infty\}$.

Those two infinity symbols are meaningless except as abstract set elements which satisfy certain conditions which are defined for the extended real number system. This system is not really well behaved in the sense of most algebraic structures. For example, it is not even closed under addition. It is an ad-hoc algebraic structure which is useful for various purposes, e.g. in measure theory or projective geometry. There are several different axiom schemes for different flavors of extended real numbers. Most often, though, these symbols are used as ad hoc symbols in analysis to indicate that a sequence or function has no finite limit, but is provably unbounded in the positive or negative direction. That's how they were used in the 19th century.

Now given the order axioms and other axioms for an extended real number system, it is generally true that $\sup R=\infty$ and $\inf R=-\infty$. But this follows almost automatically from the definition of the order relation on the extended real number set. So the equality is rigorously true, yes, but only as a consequence of the axioms. It is not true that the infinity symbols are defined to be the respective supremum or infimum. They are actually defined to be abstract elements of an extension of the real numbers, subject to specific axioms for the particular flavor of system for a given application. The equality which you give is actually a property of the axiomatic system.

Answer 2

Yes, so long as the infimum and supremum are understood to be taken in the extended reals.