Limit of Sequence to Zero Distance Point/Corollary 2

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 unbounded above, then there exists a sequence $\sequence {x_n}$ in $S$ such that $x_n \to +\infty$ as $n \to \infty$.

Proof

Let $S$ be unbounded above.

Then:

$\forall n \in \N_{>0}: \exists x_n \in S: x_n > n$

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

$\blacksquare$

Sources

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