A bounded net with a unique limit point must be convergent

Suppose, toward a contradiction, that your net $(x_i)_{i\in I}$ doesn't converge to $x$. So $x$ has an open neighborhood $N$ such that the net doesn't ultimately get into $N$. That is, if you define $J=\{i\in I:x_i\notin N\}$, then every element of $I$ is $\leq$ an element of $J$. Use that to check that $(x_i)_{i\in J}$ is a subnet of your original net $(x_i)_{i\in I}$. By compactness (of $X-N$, which is compact because $X$ is compact and $N$ is open), some subnet of $(x_i)_{i\in J}$ must converge to some point $y\in X-N$. But that subnet is also a subnet of your original net $(x_i)_{i\in I}$, so it's not allowed to converge to anything other than $x$ --- contradiction.