Yes, your proof is too informal. The first part also does not seem to be correct since you did not use the fact that is one-to-one.
Suppose is finite with elements and is one-to-one. Let Since is one-to-one, we have Therefore has elements since dsitnct give rise to distinct But also has elements and Hence therefore given any we have i.e. there exists such that i.e. is onto.
For the second part, it looks like you have the correct idea, but you need to write your proof like what I've done above. Can you do that?