I'm working through Jacobson's Basic Algebra II and a problem in the chapter on Universal Algebra is to prove that an $\Omega$-algebra $A$ is finitely-generated iff the union of any directed set of proper subalgebras is proper. It was easy to prove the $\implies$ direction, but I'm having trouble proving that the directed set condition implies $A$ is finitely-generated.
What I'm trying to do is construct a directed set $\{B_\alpha\}$ of subalgebras of $A$ where each $B_\alpha$ is finitely-generated and $A = \bigcup_\alpha B_\alpha$ so that the assumed property would imply $A = B_{\alpha_0}$ for some $\alpha_0$. This is easy enough to do if we assume a priori that $A$ is countably-generated, but I'm not sure how that can be done more generally.
Thanks Eric for the suggestion.
Take the (directed) set of all finitely-generated subalgebras $\{B_\alpha\}$ of $A$. Clearly $A = \bigcup_\alpha B_\alpha$ so that $B_{\alpha_0} = A$ for some $\alpha_0$.