Let \(\displaystyle p_n\) be the n-th prime .
Is the difference of \(\displaystyle p_{n+1}\) and \(\displaystyle p_{n}\) unlimited or not ?
Example : 3-2=1 , 23-19=4 ...etc

the maximum distance between primes certainly grows the further out you go on the prime list, but the distance between adjacent primes always has to be finite.

If it weren't then there would be a last prime which we know there is not.

......

Oh.. maybe I'm misunderstanding you.

Yes, the maximum distance between adjacent primes is an unbounded sequence, but the distance between any two adjacent primes is always finite.