Let $(X_i)_{i\ge 1}$ be a sequence of i.i.d. random variables with Bernoulli distribution of parameter $p \in (0,1)$ and with values in $\{0,1\}$.
Let $G_1 = \min \{k \ge 1 | X_k = 1\}$, $G_n = \min \{k > G_{n-1} | X_k = 1\}$ for $n \ge 2$.
So we can say that $G_n$ represents the index of the $\textrm n$'th $1$ in the random sequence. $G_n - G_{n-1}$ represents 1+#zeros between each 1.
My intuition tells me that the $(G_n - G_{n-1})_{n \ge 2}$ are i.i.d. I want to prove this. However I have trouble writing a proof of independence efficiently.
In such a situation what is a concise and formal way to prove that the $(G_n - G_{n-1})_{n \ge 2}$ are mutually independent?