Continuum Property/Corollary

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Theorem

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below.

Then $S$ admits an infimum in $\R$.

This is known as the greatest lower bound property of the real numbers.


Proof

Let $T = \set {x \in \R: -x \in S}$.

By Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above:

$T$ is bounded above.

Thus by the Continuum Property, $T$ admits a supremum in $\R$.

From Negative of Supremum is Infimum of Negatives:

$\ds -\sup_{x \mathop \in T} x = \map {\inf_{x \mathop \in T} } {-x}$

That is, by definition of $T$:

$\ds -\sup_{x \mathop \in T} x = \inf_{x \mathop \in S} x$

and so $S$ admits an infimum in $\R$.

$\blacksquare$


Sources

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