increasing sequence in a dense subset converging I have a question. Assume we have D a dense subset in $\mathbb{R}$. Then why we can always find a strictly increasing sequence in D converging to any $a\in \mathbb{R}$? Since D is dense, we can always find a sequence in D converging to a, but how we know we can always find a strictly increasing one? Thanks!

For any $c\in \mathbb R$, let $a_1\in D$ to be smaller than $c$. Then find $a_2\in (\max\\{a_1, c-\frac{1}{2}\\}, c)\cap D$, $a_3\in (\max\\{a_2, c-\frac{1}{3}\\}, c)\cap D$ etc. (By induction of course)