Taking limits of infimums. (or supremum)

Question

I'm struggling with a seemingly simple analysis step in a paper : https://projecteuclid.org/download/pdfview_1/euclid.aoap/1216677126 $\textbf{Page 1405 eq (3.11)}$. A limit of an infimum is taken (note in what follows it is taken that $\inf \emptyset=-\infty$). What is the rigorous argument to justify $$ \lim_{R\to \infty }\inf_{\{z\in \mathbb{R}^d:|z|\geq R\}}\inf_{\{f\in C(0,t) : f(t)=z \}}\int_0^t |f(s)|^2ds= -\infty ?$$