Are there any known uncountable transfinite increasing sequences of real numbers?

Question

The real numbers are uncountable, so assuming the axiom of choice there is at least transfinite sequence of real numbers $r_0, r_1, r_2, ..., r_\omega, r_{\omega + 1}, ..., $ up to (and possibly including) $r_{\omega_1}$ that contains an uncountable number of terms. However, it's not guaranteed that $r_0 < r_1 < ..., r_\omega < ...$

Is there a known transfinite sequence of real numbers that is monotonically increasing and which contains uncountably many real numbers?

Thanks!

Answer 1

If $r_{\alpha} < r_{\alpha + 1}$, then there exists a rational number between the two numbers. Hence the sequence can be corresponded with a subset of $\mathbb{Q}$, and so must be countable.