Is the statement true: Set "Omega" which is an empty set = { } is upper bounded, yet does not contain Supremum (least upper bound).

Question

So, in a lecture today we were learning what an upper bound means and how we get a supremum as the least upper bound. Yet one confusing part i saw is this statement written by the professor. Now i might be misinterpreting it but this is what it was written as, (the original was not written in english so excuse the poor terminology):

Omega = {} is upper bounded -> whichever real number we take is it's upper bound.

Sure i guess that makes sense... kind of...

Then he wrote this

Omega = {} is upper bounded, but sup({}) does not exist.

My question is, is this statement true?

I guess it makes sense, since sure any real number is "larger" than nothing. And since all real numbers are upper bounds and the set of real numbers doesn't have bounds, then we don't have a supremum.

Also, does this apply to lower bound and infimum?

I found a few similar questions to this, but they're too advanced for me right now and right now i'm learning only basic general analysis.

Thanks a lot!

Answer 1

1. $\emptyset$ is bounded from above and from below, and any real number $x$ is both an upper bound and a lower bound for $\emptyset$. This follows because in order to show that $x$ is an upper bound we have to show that for every $y\in\emptyset$ it holds that $y\leq x$, and since there are no elements in $\emptyset$ then this is vacuously true, and similarly for lower bounds.

2. Since every real number is an upper bound, this means there can be no lowest upper bound, that is, for every $x$ which is an upper bound $x-1$ is also an upper bound. This means $\sup \emptyset$ does not exist. A similar argument shows that $\inf \emptyset$ does not exist.

3. In some courses rather than saying that $\sup\emptyset$ and $\inf\emptyset$ are undefined, it is instead defined that $\sup \emptyset = -\infty$ and $\inf \emptyset = \infty$. This makes sense because the supremum is supposed to be the lowest upper bound and all real numbers are upper bounds for $\emptyset$ so it is plausible that the lowest upper bound for $\emptyset$ should be the symbol $-\infty$ which satisfies $-\infty<x$ for any $x\in\mathbb{R}$. If you didn't define it this way then don't worry about it (and don't use it). It is a useful definition to have sometimes because it can simplify some proofs, but you should make sure whether or not you defined it this way. If you didn't explicitly declare this definition then point 2 above holds instead: $\sup\emptyset$ and $\inf\emptyset$ do not exist.