Prove that $\mathbb{N}$ is homeomorphic to ${(1/n): n \ge 1}$

Question

Prove that $\mathbb{N}$ (with usual metric) is homeomorphic to $\{(1/n): n \ge 1 \}$ (with usual metric)

I have defined a function $f: n \rightarrow 1/n$ which is clearly one-one and onto. But I am not able to prove that $f$ & $f^{-1}$ is continuous in the metric space.

Thanks!

Answer 1

Because both sets are discrete (i.e., all points are isolated), every subset is open. So any function between them will be continuous. As you already have a bijection, it is a homeomorphism.