Fix a closed \(M \subseteq X\) of codimension \(\le n\) with \(\| S|_M\| \) near \(c_n(S)\). By Garling–Gordon (Theorem 7.4), for each \(\varepsilon {\gt}0\) choose a projection \(P\) with \(\ker P = M\) and \(\| P\| \le \sqrt n + \varepsilon \). Then \(L := S \circ P\) has rank \(\le \operatorname {codim} M \le n\), and \(S - L = S \circ (\mathrm{id} - P)\) has range inside \(M\) (since \(\operatorname {range}(\mathrm{id}-P) = \ker P = M\)), so \(\| S - L\| \le \| S|_M\| \, \| \mathrm{id} - P\| \le (1+\sqrt n+\varepsilon )\, \| S|_M\| \). Letting \(\varepsilon \to 0\) gives \(a_n(S) \le (1+\sqrt n)\, \| S|_M\| \); taking the infimum over \(M\) gives the claim.

Dual to the Gelfand half (Theorem 7.5). Fix \(V \subseteq Y\) of dimension \(\le n\) with \(\| \pi _V \circ S\| \) near \(d_n(S)\); by Kadets–Snobar (Theorem 7.3) choose a projection \(P\) onto \(V\) with \(\| P\| \le \sqrt n\). Then \(L := P \circ S\) has rank \(\le n\), and \(S - L = (\mathrm{id} - P) \circ S\) factors through the quotient \(Y / V\) (because \(\mathrm{id} - P\) vanishes on \(V\)), so \(\| S - L\| \le \| \mathrm{id} - P\| \, \| \pi _V \circ S\| \le (1+\sqrt n)\, d_n(S)\) after taking the infimum.

Immediate from the two halves (Theorems 7.5 and 7.6) by case analysis on which of \(c_n(S)\), \(d_n(S)\) is the smaller.