Let $H$ be a Hilbert space. Suppose that $\{u_n\} \subseteq B(H)$ is a sequence which converges to some operator $u$ in the weak operator topology, which means that for all $x,y\in H$ one has $$\langle u_n(x),y\rangle \rightarrow \langle u(x),y \rangle$$
Q1: Why must this sequence be norm-bounded?
Q2: Does it hold for nets?
Q2: Does it hold for the strong operator topology?