If $V\otimes W$ is the tensor product of vector spaces V and W, I know that for any basis $(v_i)_{i\in I}$ of V and $(w_j)_{j\in J}$ of W, $(v_i\otimes w_j)_{i\in I,j\in J}$ is a basis of $V\otimes W$, so any $a\in V\otimes W$ is a linear combination of some vectors $v_i\otimes w_j$.
But how can I prove that for every $a\in V\otimes W$ there exists $n\in \mathbb{N}$ and linear independent sets $\{v_1',...v_n'\}\in V$ and $\{w_1',...w_n'\}\in W$ such that $a=v_1'\otimes w_1'+v_2'\otimes w_2'+...+v_n'\otimes w_n'?$
This exercise is killing me: I have been trying to think of some way to construct these vectors starting with some fixed pair of bases and the upper fact, but I can't get anywhere! Help!