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

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

Answers (1)

2020-11-08
Let s be the set of negative number i.e.,
\(\displaystyle{S}={\left\lbrace{x}\in{R}:{x}\le{0}\right\rbrace}\) {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.
\(\displaystyle{u}\ge{x}\) for all x set
In set S:
\(\displaystyle{0}\ge{x}\)
\(\displaystyle\forall{x}\in{S}\)
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.
0

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