Why isn't upper bound unique?

Question

If there is an upper bound $a$ of a set $A$, and an upper bound $b$ of the set $A$ and $b>a$, why doesn't that mean that only $b$ is the upper bound of the set since it's "more upper" than $a$?

Answer 1

I think you may be confused with the terms UPPER BOUND and LEAST UPPER BOUND.

For more information on the latter term, refer to this Wikipedia page.

Answer 2

An upper bound of a set $A$, by definition, is simply

An element that is larger than (or equal to) all elements of $A$.

So, for example, if $A=[0,1]$, then $2$ is an upper bound of $A$, because we can say that $2$ is larger or equal to all elements of $[0,1]$. Also, $3$ is an upper bound of $A$, and so is $4$ and several other numbers.

By your logic, since for any $a$ which is an upper bound of $A=[0,1]$, the number $b=a+1$ is also an upper bound for $A$, then $A$ has no upper bound because no number us "more upper" than all other numbers.

Answer 3

Following the previous answers also remember "If any non empty set of real numbers has an upper bound it necessarily has a least upper bound in real numbers "