Can someone please help me to solve this problem: let $S \subset \Bbb R$ be a nonempty bounded domain. Then there exist two monotone sequences $\{x_n\}_n$ and $\{y_n\}_n$ such that $x_n, y_n \in S$ for all $n$ and $\sup S = \lim \limits _n x_n$ and $\inf S = \lim \limits _n y_n$
I tried to solve this exercise, here is what I've done: if we note $a = \sup S$ and $b = \inf S$, by the approximation property for the supremum given $n \in \Bbb N$ there exist $x_n \in S$ such that: $a-\frac{1} n \le x_n < a$, therefore by the Sandwich theorem $\lim \limits _n x_n = a$. The problem is that this sequence is not increasing.
Can someone please help me? Thanks in advance.
Your approach works well, you just need a small modification. There exist $z_n \in S$ such that $\frac {a-1} n \le z_n < a$. But instead of putting $x_n = z_n$, put $x_n = \max\{x_{n-1},z_n\}$.