If a weakly convergent net is bounded by $r$, is its limit bounded by $r$?

Question

Let $(E, | \cdot |)$ be a normed space and $E'$ its topological dual. Let $\sigma(E,E')$ be the weak topology of $E$. Let $(x_d)_{d\in D}$ be a net in $E$ that converges in $\sigma(E,E')$ to $x\in E$. Assume that $|x_d| \le r$ for all $d\in D$.

Is it true that $|x| \le r$? If $D = \mathbb N$, then the answer is clearly yes.

Answer 1

Yes, because the ball $B_r(0)$ is weakly closed (since it is closed and convex).