Order Embedding is Increasing Mapping

Theorem

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets.

Let $f:S_1 \to S_2$ be an order embedding.

Then $f$ is increasing mapping.

Proof

By definition of order embedding:

$\forall x, y \in S_1: x \preceq_1 y \implies f\left({x}\right) \preceq_2 f\left({y}\right)$

Hence $f$ is an increasing mapping.

$\blacksquare$