Ok then, prove *this*

Steve cracked that elementary proof so easily, I thought I'd throw up another interesting one - somewhat related - which I remember from my number theory class. I haven't tried to reconstruct this one yet, but I remember it being somewhat more difficult than the infinitude of primes one.

Prove that for any natural number n, there exist n consecutive composite numbers.

This really did come up in conversation

This really belongs on the lamented Fiat Lux's "e to the pi to the i" board, but since that's no longer with us, I'll use it to check whether we still have any readers here.

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Not long ago I was in Rhode Island visiting my brother and his family. Somehow the question of whether the number of primes is infinite came up in casual conversation. My sister-in-law was of the opinion that there likely is only a finite number of primes. My brother and I, both being somewhat math nerds, knew that that was contrary to fact.

"Ah-ha, but can you prove it?" Ku asked, a little smugly, knowing our best mathing years were well behind us.

It turns out I was able to recall/reconstruct the quite elementary proof in my head, right there in the kitchen. Anybody care to take a crack at it?

A couple hints:
Assume the contrary and demonstrate a contradiction.
The first step is to construct an integer with certain useful properties.