It is well known that the gap between consecutive primes is unbounded. Is this still true for a chain of consecutive primes ?

More Formally : Is the following statement true for all natural numbers m and n ?

There are m consecutive primes $a_1,...,a_m$ , such that all the gaps are greater than n (this means $a_{k+1}-a_k>n$ for all k with 1 <= k <= m-1) ?

I also heard about primes in arithmetic progressions, but I always wondered if the primes must be consecutive in such progressions.

Can any of the known properties of the prime-numbers help to answer this question ?