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.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (6)$