Can a well-ordered set isomorphic to a proper subset of itself?

Question

• Can a well-ordered set isomorphic to a proper subset of itself? Give an example or disprove it.

My answer: Yes We can. Consider $2\mathbb{N}\subseteq\mathbb{N}$. Let $f$ be the function from $\mathbb{N}$ to $2\mathbb{N}$ defined by $f(n)=2n$. We can see easily $f$ is isomorphism.

Can you check my answer? Thanks...

Answer 1

Yes, it's fine, except that you should write “let $f$ be the function…”

More generally, $\mathbb N$ is order isomorphic with every infinite subset.