The proof of this really depends upon what you have already proven and how you proved it.

What definitions and theorems you have to work with. That we don’t know.

In most developments of this material, one would normally have already have proven thatAny subset of a countable set is a countable set. Having done that what is the point of this problem?

If and is uncountable there can be no injection .

For if there were its restriction to would be an injection.

I hope you can see how different many different approaches there are and see that the one you need to use depends on what you have already done.