Existence of Prime-Free Sequence of Natural Numbers

Theorem
Let $n$ be a natural number.

Then there exists a sequence of consecutive natural numbers of length $n$ which are all composite.

Proof
Consider the number
 * $N := \left({n + 1}\right)!$

where $!$ denotes the factorial.

Then:
 * $N + 2, N + 3, \ldots, N + n, N + n + 1$

are all composite.