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Could anyone please help me figure out whether I understand this topic correctly?

It is said that if a set has an upper bound, it may have infinitely many upper bounds.

E.g. if we have a set $A = \{0,1\}$. Can I say that $2$ is an upper bound for this set as well? As there is no larger number than two in the set as well, which does not contradict to the idea that no larger number than $1$ is included in set A? $x>2$ which does not contradict $0<x<1$

$3, 4, 5 \ldots$ infinitely many numbers as upper bounds.

But, $1$, is the unique least upper bound.

Emmy
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Inna Khachikyan
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1 Answers1

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If $\alpha$ is an upper bound of a set $A$ and $\beta>\alpha$, then $\beta$ is also an upper bound of $A$. So, not only if a set has an upper bound, it may have infinitely many upper bounds, as it actually must have infinitely many upper bounds.

On the other hand, if it has a least upper bound, then it is unique.

José Carlos Santos
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