Yeah, for i in [| 2 , 100 |], 100! + i can be divided by i, hence isn't prime.
And what about the infinity of prime numbers?
For (a,b) ∈ N+*×Z,
let's define Na,b = { a*k + b, k∈Z }
= { x∈Z, x ≡ b [a] }
We consider the following topology on Z : X is an open set if X is empty or
for any element b of X there is a>0 such that Na,b is in X.
(i) any nonempty open set is infinite (easy)
(ii) the Na,b are open (easy) and closed as well, since Na,b=Z -
b-1
U Na,b+i
i=1
.
Z - {-1,1} =
U Np,0
p∈P
.
So, were P finite, Z - {-1,1} would be a finite union of closed sets (ii) ;
and consequently would be closed. Then {-1,1} would be open, which contradicts (i)... ■