$\mathbb N$ a Banach space?

Question

Is $\mathbb N$ a Banach space with the norm $|x-y|$ from $\mathbb R$? I think is Banach space because there is no convergent sequence that is not constant after some $N$. Then all limit points are in the space. But I am not sure.

Answer 1

It surely is a complete metric space, your proof is correct. Another justification of this would be that it is a closed subspace of the Banach space $(\Bbb R,|\cdot|)$. But it has no linear structure, so it's not a Banach space.

Answer 2

No, a Banach space has to be a vector space over $\mathbb R$. This is not true for $\mathbb N$ because in general $n\in \mathbb N$ does not mean that $\lambda n\in \mathbb N$ for every $\lambda \in\mathbb R$.