Proving step to full proof

I was doing a proof to prove that there exists arbitrarily long strings of composite numbers.

In more technical language, for every $\displaystyle m\in {N}$, there exists a n such that

$\displaystyle n, n+1, n+2, n+3,..., n+m$ is composite

to do so, I let $\displaystyle n=m!$,(for m>3) then we get the sequence:

$\displaystyle m!, m!+1, m!+2, m!+3,..., m!+m$

Every element of this sequence is obviously divisible by some integer, except $\displaystyle m!+1.$

Now I must prove that m!+1 is always an composite. I have no idea how to do this. I tried induction and contradiction.

Help please