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.