Unbounded Set of Real Numbers is not Compact/Normed Vector Space

Proof
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 $x_n = n$.

$\sequence {x_n}_{n \mathop \in \N}$ 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.

However:

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.