is the limit of the product of nets equal to the product of the limits of nets?

Question

Let $D$ be a directed set and $(x_\alpha)_{\alpha\in D}$, $(y_\alpha)_{\alpha\in D}$ be two convergent nets in $\mathbb{C}$. Is it true in general that the net $(x_\alpha y_\alpha)_\alpha$ is convergent and its limit equals to $\lim\limits_\alpha x_\alpha y_\alpha=\lim\limits_\alpha x_\alpha \lim\limits_\alpha y_\alpha$? If so, can you give me a hint?
Reminder: Let $(X,\tau_X)$ be a topological space and $(D,\leq)$ a directed set. A net, denoted by $(x_\alpha)_{\alpha\in D}$, is a map from $D$ to $X$.
We say that $(x_\alpha)_{\alpha\in D}$ is convergent to $x_0\in X$ iff for each $U\in \tau_X$ with $x_0\in U$ there exist $\alpha_U\in D$ such that $x_\alpha \in U$ for all $\alpha \geq \alpha_U$.