Continuum Property/Proof 2

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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 above.

Then $S$ admits a supremum in $\R$.

This is known as the least upper bound property of the real numbers.


Similarly, 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 sometimes called the greatest lower bound property of the real numbers.


The two properties taken together are called the continuum property of $\R$.


Proof

A direct consequence of Dedekind's Theorem.


$\blacksquare$