Let $[\alpha, \alpha')$ and $[\beta, \beta')$ be two intervals of ordinals such that $\alpha' - \alpha \leq \beta' - \beta$. Is there an order-preserving injection $f : [\alpha, \alpha') \rightarrow [\beta, \beta')$?
My idea would be the following. Let $i : \alpha' - \alpha \rightarrow \beta' - \beta$ be the inclusion map. Then define $f_1(x) := i(\alpha' - x) + \beta$ or $f_2(x) := \beta + i(\alpha' - x)$. However, I'm not able to show that either of them is injective.