Question

Any help is great!
Let p1,... ,pk be k different primes (not necessarily the first k primes).
The numbers that can be written as products of primes from among p1,... ,pk can be
ordered in an increasing sequence. Denote a1 < a2 < ...an < ... the terms of this sequence.
Prove that, given any integer c > 0, there is an n∈N such that an+1
−an > c.
(If k=3, p1 = 2, p2 = 5 and p3 = 7, then the sequence of the ai’s proceeds as follows:
2 < 4 < 5 < 7 < 8 < 10 < 14 < 16 < 20 < 25 < 28 < 32 < 35 < 49 < ...).