Set of Integers is not Bounded
Jump to navigation
Jump to search
Theorem
Let $\R$ be the real number line considered as an Euclidean space.
The set $\Z$ of integers is not bounded in $\R$.
Proof
Let $a \in \R$.
Let $K \in \R_{>0}$.
Consider the open $K$-ball $\map {B_K} a$.
By the Axiom of Archimedes there exists $n \in \N$ such that $n > a + K$.
As $\N \subseteq \Z$:
- $\exists n \in \Z: a + K < n$
and so:
- $n \notin \map {B_K} a$
As this applies whatever $a$ and $K$ are, it follows that there is no $\map {B_K} a$ which contains all the integers.
Hence the result, by definition of bounded space.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness