Question

Suppose that S is contained in R and that S is not closed. Is it true that there is a subsequence in S that converges to some x not in S?

Answer 1

Answer on Question #47766 – Math – Real Analysis

Question.

Suppose that $S$ is contained in $R$ and that $S$ is not closed. Is it true that there is a subsequence in $S$ that converges to some $x$ not in $S$?

Solution.

$S$ is not closed $\Rightarrow S$ contains not all its limit points $\Rightarrow \exists x \notin S, x$ is a limit point for $S \Rightarrow \exists$ subsequence $x_n \in S \ \forall n \in \mathbb{N}$: $\lim_{n \to \infty} x_n = x$.

**Example** Let $R = \mathbb{R}^1, S = (0; 2]$. Then there is a subsequence $x_n = \frac{1}{n} \in S, \lim_{n \to \infty} \frac{1}{n} = 0 \notin S$.

**Answer.** True.

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