I believe the answer to my following question is no, but some things about uncountable sets/sequences can be really counterintuitive so I wanted to double check:
Does there exist a pair of uncountable sequences $x_{\alpha}, y_{\alpha}$, indexed by $\alpha \in \mathbb{R}$ and taking values in $[0,1]$, such that $x_{\alpha} < y_{\alpha}$ for all $\alpha$ and $y_{\alpha} < x_{\beta}$ for all $\alpha < \beta$?
Would appreciate any insights!
On
Let us use another notation: $x(\alpha)$ and $y(\alpha)$. These are strictly increasing functions with image in $[0,1]$. Your assertion implies that each point of $\mathbb{R}$ is a discontinuity of $x$, but an increasing functions has at most a countable number of points of discontinuity, so this is absurd.
So you can't have this.
The same goes with image in $\mathbb{R}$
No, because this would give us an injection of $\mathbb R$ into $\mathbb Q$: for every $\alpha\in\mathbb R$, as $x_\alpha<y_\alpha$, find $q_\alpha\in\mathbb Q\cap(x_\alpha, y_\alpha)$ and notice that those $q_\alpha$ are all different: if $\alpha<\beta$ then $q_\alpha<y_\alpha<x_\beta<q_\beta$.