I am a little unsure of a proof I have written and would appreciate it being double-checked. If it is wrong and anyone has any tips to improve it, I would appreciate it. Thank you!

Prompt: "Prove that if there is a function f:A-N (set of natural numbers) that is one-to-one, then A is countable.

Proof: Let A be a set. Suppose f:A-N is one-to-one. Then, since a one-to-one function exists between A and N, A (equivalent) N. Since N is denumerable and A (equivalent) N, then A is denumerable. Thus, as A is denumerable, A is countable.

Question

I am a little unsure of a proof I have written and would appreciate it being double-checked. If it is wrong and anyone has any tips to improve it, I would appreciate it. Thank you!

Prompt: "Prove that if there is a function f:A->N (set of natural numbers) that is one-to-one, then A is countable.

Proof: Let A be a set. Suppose f:A->N is one-to-one. Then, since a one-to-one function exists between A and N, A (equivalent) N. Since N is denumerable and A (equivalent) N, then A is denumerable. Thus, as A is denumerable, A is countable.