Bounded Below Subset of Real Numbers/Examples/Real Numbers

Example of Unbounded Below Subset of Real Numbers

Let $\R$ denote the set of real numbers.

$\R$ is not bounded below.

Proof

Let $x \in \R$.

Aiming for a contradiction, suppose $\R$ is bounded below.

Then there exists $x \in \R$ such that $x$ is a lower bound for $\R$.

But then:

$x - 1 \in \R$ such that $x - 1 < x$

and so $x$ is not a lower bound for $\R$.

Hence by Proof by Contradiction $x$ is not a lower bound for $\R$.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers: Example $1.1.1 \ \text{(a)}$