If the Löwner order is a partial order, then it is transitive. If so, how can one prove it?
Proposition. Let ${\Bbb S}_n ({\Bbb R})$ denote the set of $n \times n$ symmetric matrices over $\Bbb R$. The Löwner order is transitive, i.e.,
$$ \left( \forall {\bf A}, {\bf B}, {\bf C} \in {\Bbb S}_n ({\Bbb R}) \right) \left( ({\bf A} \succeq {\bf B}) \land ({\bf B} \succeq {\bf C}) \implies {\bf A} \succeq {\bf C} \right) $$
Proof. The proposition can be rewritten as follows
$$ \left( \forall {\bf A}, {\bf B}, {\bf C} \in {\Bbb S}_n ({\Bbb R}) \right) \left( \left({\bf A} - {\bf B} \succeq {\bf O}_n \right) \land \left({\bf B} - {\bf C} \succeq {\bf O}_n \right) \implies {\bf A} - {\bf C} \succeq {\bf O}_n \right) $$
Since the addition of symmetric positive semidefinite matrices is symmetric positive semidefinite,
$$ \underbrace{(\overbrace{{\bf A} - {\bf B}}^{\succeq {\bf O}_n}) + (\overbrace{{\bf B} - {\bf C}}^{\succeq {\bf O}_n})}_{\succeq {\bf O}_n} = {\bf A} - {\bf C} \succeq {\bf O}_n \implies {\bf A} \succeq {\bf C} $$
Assuming that the proof above is correct, I am looking for alternative proofs or references.
Related: What does "curly (curved) less than" sign $\succcurlyeq$ mean?
