Let's say we have a set of numbers $\{5,9,13,17,21,\ldots, 5+4i,\ldots\}$ and each $p_i$ is a member of this set, namely $p_0 = 5, p_1 = 9, p_2 = 13$ , etc.
Consider the following functions defined recursively: \begin{align} g(p_i) &= \frac{p_i - 2}{p_i} g(p_{i-1}) & g(5) &= \frac35 \\ f(p_i) &= \frac{ p_i}2 g(p_i) && \end{align}
How does one prove that $f(p_i)$ goes to infinity as $p_i$ goes to infinity?