Forgive me if this comes off as gross abuse of maths, but it made me curious.
As the sequence of natural numbers is infinite, this would also mean there should be an infinite number of primes. However, if the distance between primes continues to grow between each new prime found, would this not mean that there would be an infinite gap between primes due to the sequence of natural numbers being infinite, meaning that there aren't an infinite number of primes?
Instead of primes, which might be difficult to visulaize and argue about, try out how well your argument holds with all the numbers consisting of a single $1$ and trailing zeroes: $1, 10, 100, 1000,\ldots$ You will see that yes, the gaps are growing endlessly (the gaps are, in order, $9, 90, 900,\ldots$), but that doesn't mean that there is a single infinite gap. Also, there are clearly infinitely many of these numbers, since for each natural number, this sequence has an entry with that many zeroes in it.
The same thing happens with primes. The distance from one prime to the next will grow (at least on average; we know that no matter how far out you go on the number line, there will still be small gaps around, but they get rare). That doesn't mean that there is an infinite gap anywhere. And it doesn't mean there are finitely many primes.