= Solution
Assume (iii), and let $\delta>0$. Choose $I$ such that
$$
\sum_{i\geq I}|\langle x_n,e_i\rangle|^2<\delta
$$
for every $n$. Coordinate convergence implies, for every finite $J\geq I$,
$$
\sum_{i=I}^J|\langle x_\infty,e_i\rangle|^2
=\lim_{n\to\infty}
\sum_{i=I}^J|\langle x_n,e_i\rangle|^2
\leq\delta.
$$
Letting $J\to\infty$ shows that the corresponding tail of $x_\infty$ is also at most $\delta$.
By the <Parseval identity for a Hilbertian basis>,
$$
\begin{aligned}
\|x_n-x_\infty\|^2
&=\sum_{i<I}
|\langle x_n-x_\infty,e_i\rangle|^2\\
&\quad+\sum_{i\geq I}
|\langle x_n-x_\infty,e_i\rangle|^2.
\end{aligned}
$$
The finite first sum tends to zero by weak coordinate convergence, while the second is at most
$$
2\sum_{i\geq I}|\langle x_n,e_i\rangle|^2
+2\sum_{i\geq I}|\langle x_\infty,e_i\rangle|^2
\leq4\delta.
$$
Since $\delta$ is arbitrary, $\|x_n-x_\infty\|\to0$. Thus (iii) implies (i), completing the equivalence and proving the <uniform basis-tail criterion for strong convergence>.
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