Here is a question, I have came up with an awnser, although im not really sure if my proof is sufficient.

Let

denote the vector space of infinite matrices with finitely many non zero enteries.

Let

be the vector space of infinite sequences in which only finitely many elements are nonzero.

Prove that

and

are isomorphic.

**My attempt at the solution:**
We know two vector spaces are isomorphic, if and only if their dimensions are equal.

Let

the finite number of non zero entires in

. This is equal to the dimension of

.

Let

the finite number of non zero elements in

. This is equal to the dimension of

Since r and m are just any finite number, we can let

, and then since the dimensions are equal, they are isomorphic.

Is this proof enough? If not a point in the right direction would be great! Thanks.