= Solution
Assume $x_n\rightharpoonup x_\infty$. For each $n$, define the bounded functional
$$
T_n(h)=\langle h,x_n\rangle.
$$
Weak convergence makes $(T_n(h))_n$ convergent, hence bounded, for every $h\in H$. The <Uniform boundedness principle> gives
$$
\sup_n\|T_n\|<\infty.
$$
By the <Riesz representation theorem>, $\|T_n\|=\|x_n\|$, so $(x_n)$ is bounded.
Taking $h=e_i$ in the definition of <weak convergence in a Hilbert space> gives
$$
\langle x_n,e_i\rangle
\longrightarrow
\langle x_\infty,e_i\rangle
$$
for every $i$. Thus (i) implies (ii).
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