Convergence of net in the Hilbert space tensor product of two Hilbert spaces

Question

Let $H$ and $K$ are two Hilbert spaces. Consider the Hilbert space $H \otimes K$, the Hilbert space tensor product of $H$ and $K$. Let $\{ x_l\}_l$ be a net in $H$ and $y$ be an element in $K$. Now consider that $\displaystyle\lim_l x_l=x$, where $x \in H$. Let us now consider the net $\{x_l \otimes y\}_l$ in $H\otimes K$. Now I want to show that, the given net $\{x_l \otimes y\}_l$ converges to $x \otimes y$ in $H \otimes K$.
I know that, the norm of tensor product is defined by $\|x \otimes y\|=\|x\|\|y\|$. But I am unable to show that $\displaystyle\lim_l (x_l\otimes y)=x\otimes y$. Please help me to solve this. Thank you for your time and help.

Answer 1

$$\|x_l\otimes y- x\otimes y\| = \|(x_l-x)\otimes y\| = \|x_l-x\|\|y\| \stackrel{l\to \infty}\longrightarrow 0.$$