Set of Integers is not Bounded

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Let $\R$ be the real number line considered as an Euclidean space.

The set $\Z$ of integers is not bounded in $\R$.


Let $a \in \R$.

Let $K \in \R_{>0}$.

Consider the open $K$-ball $\map {B_K} a$.

By the Archimedean Principle 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.