LET $(a_n : n = 0, 1, 2,\dots)$ be a sequence of distinct elements of a totally ordered set $X$. Can the sequence necessarily be rearranged so as to form a strictly increasing sequence?
Certainly this is possible if $X$ is in fact well-ordered: Define recursively $b_0 = \min \{a_n : n = 0, 1, 2,\dots\}$, $b_1 = \min \{a_n : n = 0, 1, 2,\dots\} \setminus \{b_0\}$, $b_2 = \min \{a_n : n = 0, 1, 2,\dots\} \setminus \{b_0, b_1\}$, etc.
How about $X=\mathbb{R}$ and the sequence $a_n=\frac{1}{n+1}$? Since it doesn't have the smallest number, we can't even find $b_0$.