Solvable Number Field Extensions of Bounded Root Discriminant

Let $K$ be a number field and $d_K$ the absolute value of the discrimant of $K\mathrm{/}\mathbb{Q}$. We consider the root discriminant $d_L^{\frac{1}{[L:\mathbb{Q}]}}$ of extensions $L\mathrm{/}K$. We show that for any $N>0$ and any positive integer n, the set of length n solvable extensions of $K$ with root discriminant less than $N$ is finite. The result is motivated by the study of class field towers.