I read that, with the Scott topology, suprema of sequences are topological limits (See page 1 of this article).

Let $(X, \le)$ be a DCPO, and $D$ be a directed subset of $X$. I can easily see that the identity net on $D$ converges to $\bigvee D$.

How about the other direction? If the identity net on $D$ converges to $x$, does it imply that $x = \bigvee D$?