= Solution
Let
$$
C_n=\overline{W(Q_nAQ_n^*)}.
$$
The tail spaces decrease, so $C_{n+1}\subseteq C_n$. If $\lambda\in\bigcap_nC_n$, choose a unit vector $v_n$ supported after coordinate $n$ with $|\langle Av_n,v_n\rangle-\lambda|<1/n$. Such vectors converge weakly to zero, hence $\lambda\in W_e(A)$.
Conversely, if $v_j\rightharpoonup0$ and $\langle Av_j,v_j\rangle\to\lambda$, then for every fixed $n$ the first $n$ coordinates of $v_j$ tend to zero. After normalizing $Q_nv_j$, its numerical values still tend to $\lambda$, so $\lambda\in C_n$. Therefore
$$
\boxed{\bigcap_{n=1}^\infty C_n=W_e(A)}.
$$
When the intersection is nonempty, decreasing closed sets have distance functions increasing pointwise to the distance from their intersection; on each compact set this convergence is uniform. This is precisely
$$
\boxed{C_n\downarrow W_e(A)
\quad\text{in the <Attouch--Wets topology>}}.
$$
If $W_e(A)=\varnothing$ and a compact $K$ met every $C_n$, nestedness and compactness would supply a convergent sequence whose limit belongs to all $C_n$, a contradiction. Hence $K\cap C_n=\varnothing$ for all sufficiently large $n$.
Solved by gpt-5.6-sol high.
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