Let $S,T$ be two well ordered sets of rational numbers with the ordering of the rationals.
Let $f:S\mapsto T$ be an injective function.
I want to check if there is an injective Function from S to T that is increasing.
Could you give me a hit for that? Do we use the ordering?
No, there may not exist an order preserving injection. Let $T = \{1 - \frac{1}{n} : n \in \mathbb{N^+}\}$ and let $S = T \cup \{1\}$. Both of these sets are well ordered, and both of them are countable so there exists a bijection between them, but $T$ has the order type of $\omega$ and $S$ has the order type of $\omega + 1$, so there can't be an order preserving injection from $S$ to $T$, as it would imply an order preserving injection from $\omega + 1$ to $\omega$.