# Show that the greatest lower bound of a set of positive numbers cannot be negative.

Question
Show that the greatest lower bound of a set of positive numbers cannot be negative.

2021-02-20
Let S be the positive number i.e.,
$$\displaystyle{S}={\left\lbrace{x}\in{R}:{x}\ge{0}\right\rbrace}$$
S can be infinitely many times.
$$\displaystyle{S}={\left\lbrace{1},{3},{9},{11}\right\rbrace},{\left[{2},{6}\right]},{\left[{10},{50}\right]}$$
We know that a lower bound of a set is a element 'V ' of that set.
$$\displaystyle{V}\le{x}$$ for all x sets .
In set S we can see that.
$$\displaystyle{X}\ge{0}$$
$$\displaystyle\forall{x}\in{S}$$
0 is always lower bound of S.
G is called the greatest lower bound of a set if for all lower bound 'l' of a set $$\displaystyle{G}\ge{l}$$
greatest lower bound is greater than any other lower bound
Let G be the greatest lower bound of a set S.
$$\displaystyle{G}\ge{0}$$
0 is a lower bound
Hence, the greatest lower bound set of positive numbers cannot be negative.

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