No isomorphism between set of finite and infinite sequences

Question

I have this in my notes and now I can't remember why that's the case:
If $V$ is the set of finite sequences (i.e. finitely many non-zero entries) and $V'$ is the set of infinite sequences, then $V$ is not isomorphic to $V'$.
I was thinking along the line that the first one has a countable basis (1 in entry n and zero everywhere else) and the second one doesn't, but I'm not sure if this works and also can't formalise it.
I really appreciate your help!