Unbounded Set of Real Numbers is not Compact

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Let $\R$ be the set of real numbers considered as a Euclidean space.

Let $S \subseteq \R$ be unbounded in $\R$.

Then $S$ is not a compact subspace of $\R$.

Proof 1

By the rule of transposition, it suffices to show that if $S$ is a compact subspace of $\R$, then $S$ is bounded.

Let $\CC$ be the set of all open $\epsilon$-balls of $0$ in $\R$:

$\CC = \set {\map {B_\epsilon} 0: \epsilon \in \R_{>0}}$

We have that:

$\ds \bigcup \CC = \R \supseteq S$

From Open Ball of Metric Space is Open Set, it follows that $\CC$ is an open cover for $S$.

Let $\FF$ be a finite subcover of $\CC$ for $S$.

Then $\ds \bigcup \FF$ is the largest open $\epsilon$-ball in $\FF$.


$S \subseteq \ds \bigcup \FF \in \CC$

Hence, $S$ is bounded.


Proof 2


the result follows by the rule of transposition.


Proof 3

We have that $\struct {\R, \size {\, \cdot \,} }$ is a normed vector space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence such that $x_n = n$.

Aiming for a contradiction, suppose $\sequence {x_n}_{n \mathop \in \N}$ poseses a convergent subsequence $\sequence {x_{n_k}}_{k \mathop \in \N}$.

By Convergent Sequence is Cauchy Sequence, $\sequence {x_{n_k}}_{k \mathop \in \N}$ is Cauchy.


\(\ds \forall k, m, n \in \N: \, \) \(\ds \size {x_{n_k} - x_{n_m} }\) \(\ge\) \(\ds \size {x_{n_{m + 1} } - x_{n_m} }\)
\(\ds \) \(\ge\) \(\ds \size {x_{n + 1} - x_n }\)
\(\ds \) \(\ge\) \(\ds \size {n + 1 - n}\)
\(\ds \) \(=\) \(\ds 1\)

This contradicts the definition of Cauchy sequence.

Hence, there is no convergent subsequence of $\sequence {x_n}_{n \mathop \in \N}$.

By definition, $\struct {\R, \size {\, \cdot \,} }$ is not compact.