Can a Natural Number Satisfy n ≡ 1 (mod p) for All Primes in a Large Set?

[Bibubo]

Let $\left\{ p_{1},p_{2},\dots,p_{h}\right\}$ a set of consecutive prime numbers. I want to show that, if $h$ is large enough, then doesn't exists a natural number $n$ such that $$n\equiv1\textrm{ mod }p_{i},\,\forall i=1,\dots,h.$$
I think is true but I have no idea how to prove it. Am I wrong?

 

[Nono713]

Your claim is unfortunately false. Try $n = p_1 p_2 \cdots p_h + 1$ (and the primes don't need to be consecutive either).