A bounded net with a unique limit point must be convergent Let $(x_i) _{i \in I}$ be a net in a compact Hausdorff space X, with the property that every convergent subnet has the same limit $x$. Does this imply that $(x_i) _{i \in I}$ converges?
If I try to proceed by contradiction, I do not how to rule out the possibility of having subnets that do not converge at all. (Of course, by compactness each such subnet should have at least a subnet convergent to $x$, but this does not help me.)
(If it helps, my compact space is in fact a weakly-compact subset in a locally-convex linear topological space space, obtained with Alaoglu's theorem.)

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.
