Take a maximising sequence \(x_n\) of unit vectors with \(\| S x_n\| \to \| S\| \); by compactness extract a subsequence with \(S x_{n_k} \to y\), and set \(v := \| S\| ^{-1} y\), \(u' := S^* v\). Cauchy–Schwarz on \(\langle u', x_{n_k}\rangle = \langle v, S x_{n_k}\rangle \to \| S\| \) gives \(\| u'\| = \| S\| \); equality in Cauchy–Schwarz forces \(x_{n_k} \to u\), so by continuity \(S u = y = \| S\| v\).

By iterating Theorem 6.4: peel off the top singular pair \((\sigma _0, u_0, v_0)\) of \(S\), replace \(S\) by \(S - \sigma _0 \langle u_0, \cdot \rangle v_0\) (which is again compact), and recurse. The variational characterisation of each \(u_k\) forces the \(u_k\) (and dually the \(v_k\)) to be orthonormal; \(\sigma _k \to 0\) follows from compactness; and the partial sums converge to \(S\) in operator norm because the \(n\)-th residual has norm \(\sigma _n\). Eckart–Young (Theorem 6.6) identifies \(\sigma _k = a_k(S)\).

Take the truncated SVD \(L = \sum _{k {\lt} n} a_k(S)\langle u_k, \cdot \rangle v_k\) from Theorem 6.5, which has rank \(\le n\). The residual \(S - L = \sum _{k \ge n} a_k(S)\langle u_k, \cdot \rangle v_k\) acts on the orthonormal \((u_k)_{k \ge n}\) with weights \(a_k(S) \le a_n(S)\), so its operator norm is exactly \(a_n(S)\); no rank-\(\le n\) operator does better, since \(a_n(S)\) is by definition the infimum of \(\| S - L'\| \) over such \(L'\).

Using the top \(n+1\) singular pairs of Theorem 6.5, let \(A \in \mathcal{L}(\ell _2^{n+1}, H_1)\) send the basis vector \(e_k\) to the right singular vector \(u_k\), and let \(B \in \mathcal{L}(H_2, \ell _2^{n+1})\) send \(y\) to \(\sum _{k \le n} \langle v_k, y\rangle e_k\), i.e. project onto the span of the left singular vectors. Both are contractions because \((u_k)\) and \((v_k)\) are orthonormal. Then \(S u_k = a_k(S)\, v_k\) gives \((B \circ S \circ A) e_k = a_k(S)\, e_k\) for \(0 \le k \le n\).

Since \(c {\lt} a_n(S)\), no rank-\(\le n\) operator approximates \(S\) to within \(c\), so there is an \((n+1)\)-dimensional subspace \(M \subseteq H_1\) on which \(\| S x\| \ge c\, \| x\| \). Let \(A\) embed \(\ell _2^{n+1}\) isometrically onto \(M\); then \(S \circ A\) is bounded below by \(c\), so \(c\, (S \circ A)^{-1}\) is a contraction on its range, and composing it with the orthogonal projection onto that range gives a contraction \(B\) with \(B \circ S \circ A = c \cdot \mathrm{id}_{\ell _2^{n+1}}\). No compactness is used.