TITLE: Must a weakly or weak-* convergent net be eventually bounded?
QUESTION [25 upvotes]: Let $\mathfrak{X}$ be a Banach space.  As a standard corollary of the Principle of Uniform Boundedness, any weak-* convergent sequence in $\mathfrak{X}^*$ must be (norm) bounded.  A weak-* convergent net need not be bounded in general, but must it be eventually bounded?  
It seems like the following should prove that the answer is yes: If $\{y_\nu\}$ is a net in $\mathfrak{X}^*$, suppose it's not eventually bounded.  Then we can recursively construct an unbounded subsequence: since the net is not bounded, there exists some $\nu_1$ with
$\|y_{\nu_1}\| > 1$.  By hypothesis the tail subnet $\{y_\nu \mid \nu \geq \nu_1\}$ is not bounded, so there exists some $\nu_2 \geq \nu_1$ with $\|y_{\nu_2}\| > 2$, and so on.  If the original net were weak-* convergent, so would this unbounded subsequence, contradicting PUB.
It would then follow that weakly convergent nets in $\mathfrak{X}$ are bounded as well, because the image in $\mathfrak{X}^{**}$ would be weak-* convergent.
Question: This is legit, right?  I'm still not quite comfortable enough with nets or with the weak-* topology to entirely trust myself here, and I'd like to know the answer since I seem to be bumping into this question a lot recently.

REPLY [13 votes]: Nate Eldredge has done the hard work by giving a counterexample to the conjecture; here’s a brief explanation of what’s wrong with the argument given in the question.
A net $\psi:J\to X$ is a subnet of a net $\varphi:I\to X$ iff for each $i\in I$ there is a $j\in J$ such that $$\big\{\psi(j\,'):j\le j\,'\big\}\subseteq\big\{\varphi(i\,'):i\le i\,'\big\}\;.$$ Equivalently, if $\varphi$ is eventually in a set $A$, so is $\psi$. 
Taking $D$ as the directed set underlying your net, there’s no reason to think that your sequence $\langle y_{\nu_k}:k\in\Bbb N\rangle$ is actually a subnet of $\langle y_\nu:\nu\in D\rangle$: there may well be a $\nu_0\in D$ such that $$\{y_{\nu_k}:k\in\Bbb N\}\setminus\{y_\nu:\nu_0\preceq\nu\}$$ is infinite. This is the case with Nate’s net, for instance.