Let $K$ be a complete discrete valuation field, $K_\infty/K$ an infinite extension which is a directed union of finite extensions $\{K_i/K\}$. Let $L$ be a finite extension over $K$ and disjoint from $K_\infty$ over $K$. Let $L_i=LK_i$, similarly for $L_\infty$. Prove that the residue field degree $[k_{L_\infty}:k_{K_\infty}]=[k_{L_i}:k_{K_i}]$ for all large $i$.
I want to show for a large $i$, $k_{L_j}=k_{K_j}k_{L_i}$ for all $j \geq i$ and $\infty$, then I can show above statement. But I don’t know whether this is true.