My professor asked us to think about this question: If $n$ is a finite cardinal number, what is the value of $n \aleph_0$?
I think that the answer is $\aleph_0$ again. We had a similar property in addition. If $\aleph_0$ is the cardinal number of the natural numbers and $n = card \{x_1, x_2,...,x_n\}=X$ then $n\aleph_0$ is the cardianl number of the cartesian product of $X$ and the set of natural numbers. Then to prove my guess I need a function that is surjective and injective from the natural numbers to $X \times \Bbb N$. Right? Can someone give me a hint how to find such function? (If my guess is right) Or is there a better approach to solve such questions?