Show that the least upper bound of a set of negative numbers cannot be positive.

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Answer 1

Let s be the set of negative number i.e.,

S = {x \in R : x \leq 0}

{x be any negative number}
The least upper bound of s cannot be positive.
We know that an upper bound of a set is an element say u belongs to that set.

u \geq x

for all x set
In set S:

0 \geq x

\forall x \in S

0 is an upper bound of s
S is not a set of all real numbers it is a set which contain negative numbers only.

Show that the least upper bound of a set of negative numbers cannot be positive.

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