Let $X \subset \mathbb R$ be a seq. $\{x_n\}$ in $\mathbb R$ that is dense in $\mathbb R.$ What is the set of limit points of $\{x_n\}$ ?
Answer part:
So we know that $X \cap (x_1, x_2) \neq \emptyset $ such that $x_1 < x_2$ from the dense definition. I guess it should be an increasing sequence. What to do next ? Is it to define a subsequence from $x_1's$, $x_2's$ etc. then show that its partial limit is indeed a infinite set of ($x_1$, $x_2$, ... ) ?
Can you help me with this ?
Thanks...