We have $\aleph_1\leq |\mathbb{R}|$. Do we know if there exists an increasing $\aleph_1$ sequence of real numbers? (That is, a set $\{a_\theta\in\mathbb{R}:\theta<\omega_1\}$ such that $a_{\theta_1}<a_{\theta_2}$ if $\theta_1<\theta_2$.) If we know there exists such a sequence, is it possible to have some explicit example/construction?
This question arose because it was very easy for me to construct an increasing $\aleph_0$ sequence of real numbers, but once I thought about increasing $\aleph_1$ sequence, I couldn't even confirm that such a sequence exists. For an increasing $\aleph_0$ sequence, we can take $\{-\frac{1}{n}:n\in\mathbb{N}\}$, or just $\mathbb{N}$ it self.