Prime-bounded subwords

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In the number 373 all subwords (3, 7, 37, 73, and 373) are prime. Similarly, in 9719 all subwords are divisible by at most one prime. And similarly again in 7319797913 all subwords are divisible by at most two primes. These are the largest integers with their respective properties. We show for any k≥ 1 there are only finitely many integers having subwords divisible by at most k primes. In fact, we show for any B and d coprime that n contains a base-B subword divisible by d if n>Bᵈ. So as example consequence, past a certain point every prime contains a subword divisible by say 10000000007.