Exercise
I'm a little stuck on an Exercise in Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range. The Exercise (E.II.5.1) is as follows (here $\mathcal{O}(D)$ is the family of holomorphic functions on $D$):
Let $D\subset\mathbb{C}$ and suppose $f_{v}\in\mathcal{O}(D)$ for $v=0,1,2,\ldots$. Define the Hartogs regularity radius $R(z)$ of the series \begin{equation} \sum_{v=0}^{\infty}f_{v}(z)w^{v}, w\in\mathbb{C}, \end{equation} by setting $R(z)$ equal to the radius of convergence of the above in $w$. Prove that $-\log R(z)$ is subharmonic on $D$. (This is fundamental result of Hartogs, which is at the core of Oka's Theorem 5.6.)
My attempt
Note that, explicitly, $$ R(z)=\frac{1}{\limsup_{v\to\infty}\sqrt[v]{|f_{v}(z)|}} $$ so \begin{equation} -\log R(z)=\log\left(\limsup_{v\to\infty}\sqrt[v]{|f_v(z)|}\right). \end{equation} Let $\{v_j\}$ be a subsequence such that $$ \lim_{j\to\infty}\sqrt[v_j]{|f_{v_j}(z)|}=\limsup_{v\to\infty}\sqrt[v]{|f_{v}(z)|} $$ and define a sequence $$ u_j(z):=\log\left(|f_{v_j}(z)|^{1/v_j}\right). $$ Clearly $\lim_{j\to\infty}u=-\log R(z)$, and (as has been shown) each $\log|f_{v_j}(z)|$ is subharmonic. Hence each $ u_j(z)=\tfrac{1}{v_j}\log |f_{v_j}| $ is subharmonic.
From here, I would like this show that this implies the limit function is subharmonic, but I don't know how to go about it. One way I know could do it by showing the sequence $\{u_j\}$ is decreasing, but I don't believe this is the case.
Also, this is the first exercise in the set for this section, so I feel as though there might be a significantly easier way. If anyone has any hints/tips on how to repair this argument or possibly have an idea on a simpler one, it would be a huge help. Thanks in advance.