if $y,z$ converges to zero, does $yz \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$ as well?

Question

Let $\Omega=(0,1)$ and $y,z \in L^2(\Omega)$ such that $$y \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$$ $$z \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$$

can we deduce that $yz \rightarrow 0 \,\, \text{in} \, L^2(\Omega)$ ?

Answer 1

No. Can you find $y \in L^{2}$ such that $y^{2} \notin L^{2}$? Once you have such a $y$ take $y_n=z_n=\frac y n$.