How to prove a bound function for a sequence of numbers?

Question

Let $G_n$ be defined by

$$G_n = \begin{cases} 1 & n=0 \\ 2 & n = 1 \\ 3 & n = 2 \\ 4 & n = 3 \\ 2G_{n-1}-2G_{n-3}+G_{n-4} & n\geq4 \end{cases}$$

How can I prove that $f(n) = n$ is a bound function (or loop variant) for the above sequence?