Limit of Sequence to Zero Distance Point/Corollary 1

Theorem
Let $S$ be a non-empty subset of $\R$.

If $S$ is bounded above, then there exists a sequence $\left \langle {x_n} \right \rangle$ in $S$ such that $\displaystyle \lim_{n \to \infty} x_n = \sup S$.

If $S$ is unbounded above, then there exists a sequence $\left \langle {x_n} \right \rangle$ in $S$ such that $\displaystyle x_n \to +\infty$ as $n \to \infty$.

Proof
If $\xi = \sup S$, then from Distance from Subset of Real Numbers, $d \left({\xi, S}\right) = 0$.

The result then follows directly from Limit of Sequence to Zero Distance Point.

Note that the terms of this sequence do not necessarily have to be distinct.

If $S$ is unbounded above, then:
 * $\forall n \in \N_{>0}: \exists x_n \in S: x_n > n$

Hence $x_n \to +\infty$ as $n \to \infty$.