I have to prove that if $A \cap B \neq \emptyset$ then $\inf(A\cap B) \geq \max\{\inf(A,\inf(B)\}$
The proof in my lecture notes is the following
However I developed a much shorter proof and I don't think it is actually necessary to prove it that way. Do you think mine works as well?
Without loss of generality suppose $\inf(A)\geq \inf(B)$ so that $\max\{\inf(A),\inf(B)\}=\inf(A)$.
Take an arbitrary $c\in A\cap B$. By definition $c\in A$ and $c\in B$. Hence $c\geq \inf(A)\geq\inf(B)$. So that $\inf(A)$ and $\inf(B)$ are lower bounds for $A\cap B$. Therefore by definition of infimum $\inf(A\cap B) \geq \inf(A)\geq\inf(B)$ and in particular $\inf(A \cap B) \geq \max\{\inf(A),\inf(B)\}$. The same holds if the first assumption is inverted.