The fact that if I have two extensions of number fields, say $L_1 / K$ and $L_2 / K$, unramified at a discrete valuation $v$, then the compositum $L_1L_2 / K$ is unramified at $v$ can be proven directly. I'm now in a situation where I would like to take the compositum over infinitely many extensions $L_i / K$ each of which I know is unramified at $v$.
Knowing the result for two extensions can be used directly to see that the result holds for finitely many extensions. Is there an easy way to see that the result holds for infinitely many extensions as well? In my case, the compositum might have infinite degree over $K$.