Although I have read that it's quite easy to prove there isn't an $\omega_1$ increasing sequence on real set I spent a lot of time figuring out why it happens and finally I think I made it, but I'm not sure about it. Here is my approach.
First of all, I supossed there was an increasing sequence $s: \omega_1\longrightarrow\mathbb{R}$ and I considered the following sets
$D_n=\{\alpha<\omega_1\;\vert\;s(\alpha+1)-s(\alpha)<\frac{1}{n}\}$
After that, to demonstrate that there was a set $ D_n $ such that its cardinality was $\aleph_1 $, I assumed not, but that would mean that all sets $ D_n $ were countable and then their union would also be countable so that there is $ \aleph_1 $ ordinals less than $ \omega_1 $ so that each of them does not belong to any set $ D_n $ (due to the regularity of $ \omega_1 $), that would mean that for all of them $ s (\alpha + 1 ) -s (\alpha) \geq 1 $ messing up the injection of $ s $ because there would be $ \aleph_1 $ ordinals in the domain and $ \aleph_0 $ in the range.
Finally, the existence of a $D_{n_{0}}$ set such that its cardinality is $\aleph_1$ is a contradiction for the following reason. Take a random $\alpha$ in $D_{n_{0}}$, then $s(\alpha+1)-s(\alpha)<\frac{1}{n_0}$ but also, by the archimedean postulate, there would be a natural number $m$ such that $\frac{1}{m}<s(\alpha+1)-s(\alpha)<\frac{1}{n_0}$ and that number $\frac{1}{m}$ would never reach $\frac{1}{n_0}$ ,because there are $\aleph_1$ numbers between them, and that would contradicts the archimedean postulate.
Am I right up to this point?
Thanks in advance for your help and time.
PD: I'm a begginer Set theory student so sorry if I said something that didn't make sense.
I’m afraid that I don’t understand your argument that no $D_n$ can be uncountable; in particular, I can’t make any sense of the assertion that ‘$\frac1m$ would never reach $\frac1{n_0}$’.
In any case, you’re working much too hard: all you have to do is note that each interval $\big(s(\alpha),s(\alpha+1)\big)$ must contain a rational number $q_\alpha$, and since these intervals are pairwise disjoint, $\{q_\alpha:\alpha\in\omega_1\}$ would then be an uncountable set of rational numbers, which is absurd.