The reals are unbounded above.

by RoRi

June 30, 2015

Prove that there is no $a \in \mathbb{R}$ such that $x \leq a$ for all $x \in \mathbb{R}$ .

Proof.
Let $a \in \mathbb{R}$ be given. Then, since $0 < 1$ (Theorem I.21) we have $0+a < 1+a \implies a < 1+a$ for all $a \in \mathbb{R}$ (Theorem I.18). Then take $x = a+1$ and we have $x > a$ ; hence $x \not \leq a$ . Hence, there is no $a \in \mathbb{R}$ such that $x \leq a$ for all $x \in \mathbb{R}. \qquad \blacksquare$

Stumbling Robot

The reals are unbounded above.

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