Hi

I know that this is a basic question but it is one that I cannot seem to be able to get my head around. If I have a set

and another set where the thing condition that there can only be finitely many x_k different from zero is replaced by the condition that the supremum of x_k must be finite.

How can I show that the second set is a subspace of the first set?

I know about the addition and multiplication with scalar, but I just cannot seem to be able to understand it so I was hoping for a thorough example.

Thanks a lot