Dedekind's Theorem/Corollary

Theorem

Let $\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\R$.

Then either $L$ contains a largest number or $R$ contains a smallest number.

Proof

From Dedekind's Theorem, there exists a unique real number such that:

$l \le \gamma$ for all $l \in L$

$\gamma \le r$ for all $r \in R$.

Let $\gamma \in L$.

Then by definition $\gamma$ is the largest number in $L$.

Let $\gamma \in R$.

Then by definition $\gamma$ is the smallest number in $R$.

By the definition of Dedekind cut, $\tuple {L, R}$ is a partition of $\R$.

Hence $\gamma$ is either in $L$ or $R$, but not both.

That is, $\gamma$ is either:

the largest number in $L$

or:

the smallest number in $R$.

Hence the result.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.32$. Corollary