Poisson and central limit theroem

Hi, I am having trouble understanding this logic:

If we have n independent Poisson random variables each with a parameter 1/n. Then, the sum of these random variables is a Poisson with parameter 1.

According to the central limit theorem, as n goes to infinity, the distribution of the sum goes to a normal distribution. But in this case, the distribution of the sum still goes to a Poisson with parameter 1, which is not a normal distribution.

Why is there such a contradiction?

Thanks!