Strictly increasing function from $\alpha< \aleph_1$ to $\mathbb{R}$

Question

I know that there is no increasing function $f: \aleph_1 \to \mathbb{R}$, so it seems like for $\alpha < \aleph_1$, there should exists a function $f: \alpha \rightarrow \mathbb{R}$ that is strictly increasing.

I can't think of how this could be shown/constructed. Any ideas?