Suppose one finds the earliest prime gap of at least a certain size $g$, so that $p_{n+1}-p_n=g$ and $n$ is the smallest index for which the gap is as big as $g$.
Now consider the relative size of the gap: $\dfrac{p_{n+1}-p_n}{p_n}$
When will $n$ be the largest index for which the relative size is that big?
Is the set of all values of $n$ for which that happens infinite?
Is the set of all values of $n$ for which that does not happen infinite?
(For example, this happens when $p_n=3$ and when $p_n = 113$.)