Real analysis, topology

Question

"If a set is equivalent to one of its proper subset then it is infinite set"

I was wondering why can't it be countably infinite? Since $\mathbb{Z} \sim \mathbb{N} \sim \mathbb{Z}+ \rightarrow \mathbb{Z}+ \sim \mathbb{Z}$, $\mathbb{Z}$ is countably infinite hence $\mathbb{Z}$ is equivalent to its proper subset but Z is countably infinite.. Please correct me where I am getting it wrong

Answer 1

There is no contradiction. Countable infinity is just a special kind of infinity.