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Assume (iii), and let $\delta >0$. Choose $I$ such thatfor every $n$. Coordinate convergence implies, for every finite $J\ge I$,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,The finite first sum tends to zero by weak coordinate convergence, while the second is at mostSince $\delta$ is arbitrary, $\Vert x_n-x_{\infty }\Vert \to 0$. Thus (iii) implies (i), completing the equivalence and proving the uniform basis-tail criterion for strong convergence.