# Would it be true that if a set is bounded

## Main Question or Discussion Point

In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?

HallsofIvy
Homework Helper

In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?
Define your terms! A "set" does not even have to consist of numbers. In order to talk about a set being "bounded" there must be some kind of metric defined on it but even then there may not be an order- the set of all complex numbers with norm less than 1 is bounded but is not an ordered set and so "supremum" makes no sense.

I assume you are talking about a set of real numbers. Yes, any set of real numbers, having an upper bound, has a real number as supremum. That's one of the "defining" properties of the real numbers.

It is NOT true, for example, if you are talking about rational numbers. For example, the set of all rational numbers, x, such that $x^2< 2$ has such numbers as 1.5, 2, etc. as upper bounds but does not have a rational number as supremum. As a set of real numbers, its supremum is $\sqrt{2}$.