Limit of Sequence to Zero Distance Point/Corollary 2
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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 $\S 4.29 \ (6)$