= Solution
Let $P_n:\ell^2(\mathbb N)\to\mathbb C^n$ be coordinate projection and set
$$
T_n=P_nAP_n^*.
$$
This finite matrix is computable from the matrix entries of $A$. Compute its <singular value decomposition>
$$
T_n=V_n\Sigma_nW_n^*
$$
and define the <unitary polar factor of a finite compression>
$$
\boxed{A_n=V_nW_n^*.}
$$
This is a unitary operator on $\mathbb C^n$, including when $T_n$ is singular.
Put $Q_n=P_n^*P_n$. Since $Q_n\to I$ strongly and $A$ is unitary,
$$
P_n^*T_n^*T_nP_nv
=Q_nA^*Q_nAQ_nv\longrightarrow v.
$$
The continuous functional calculus for positive matrices therefore gives
$$
P_n^*|T_n|P_nv\longrightarrow v.
$$
The polar identity $T_n=A_n|T_n|$ now yields
$$
\|P_n^*(A_n-T_n)P_nv\|
=\|(I-|T_n|)P_nv\|\longrightarrow0.
$$
Also $P_n^*T_nP_n=Q_nAQ_n\to A$ strongly, and hence
$$
\boxed{P_n^*A_nP_n\longrightarrow A\quad\text{strongly}.}
$$
In particular the weak convergence required in part (c) holds. The construction uses only a finite block of the given matrix and a finite <singular value decomposition>, so it is an algorithm realizing all the assumptions of part (c).
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