Query about another post $\operatorname{rank}(AB) \leq \min(\operatorname{rank}(A), \operatorname{rank}(B))$

Question

For, $\operatorname{rank}(AB) \leq \min(\operatorname{rank}(A), \operatorname{rank}(B))$

On this post I don't understand why it is enough to prove $\dim \operatorname{range}(AB)\leq \dim R(A)$ and $\operatorname{range}(AB)\leq \dim R(B)$. Wouldn't the "min" part play a role as well?

Answer 1

Proving both $a\le b$ and $a\le c$ is the same as proving $a\le\min\{b,c\}$.