Limit of Sequence to Zero Distance Point/Corollary 1

Corollary to Limit of Sequence to Zero Distance Point
Let $S$ be a non-empty subset of $\R$.

Let the distance $\map d {\xi, S} = 0$ for some $\xi \in \R$.

If $S$ is bounded above, then there exists a sequence $\sequence {x_n}$ in $S$ such that:
 * $\ds \lim_{n \mathop \to \infty} x_n = \sup S$

Proof
Let $\xi = \sup S$.

Then from Distance from Subset of Real Numbers:
 * $\map d {\xi, S} = 0$

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.