I have two questions which are intuitively true.

Let $V$ be a Hilbert space. As usual we can turn $V\otimes V$ or $V\otimes V\otimes V$ into Hilbert spaces by intorducing the natural inner product and by performing completion.

Question 1. We have a sequence of simple tensors $f_{i}\otimes g_{i}$ that converges in the Hilbert space $V\otimes V$ to some tensor. Is it true that the limit is a simple tensor, that is can be represented by $f\otimes g$? The matter is that $f_{i}$ and $g_{i}$ need not have limits, as the simple example $i f\otimes\frac{1}{i} g$, for some fixed $f,g\in V$ shows.

Question 2. If a sequence of simple tensors of the form $f_{i}\otimes f_{i}\otimes g_{i}$ has a limit which is a simple tensor, can it be represented by $f\otimes f\otimes g$, for some $f,g\in V$?