Axioms (S1)–(S4) follow directly from the definition of \(a_n\) as an infimum: monotonicity and nonnegativity are inherited from the underlying set of approximants, subadditivity (S2) uses that \(A\) approximates \(S + T\) whenever it approximates \(S\) up to \(\| T\| \), the ideal property (S3) uses \(\operatorname {rank}(B A' A) \le \operatorname {rank}(A')\), and (S4) is witnessed by the approximant \(A = S\). For (S5\('\)) let \(L\) have \(\operatorname {rank}(L) \le n {\lt} \dim X\); then \(\operatorname {range}(L)\) is a proper finite-dimensional, hence closed, subspace of \(X\), and Riesz’s lemma provides, for every \(r {\lt} 1\), a unit vector \(x_0\) with \(\| x_0 - L x_0\| \ge r\), so \(\| \mathrm{id} - L\| \ge r\); letting \(r \to 1\) and combining with \(a_n(\mathrm{id}) \le \| \mathrm{id}\| = 1\) gives \(a_n(\mathrm{id}_X) = 1\). The classical (S5) on \(\ell _2^{n+1}\) is the special case \(X = \ell _2^{n+1}\). The argument needs neither the SVD nor Auerbach’s lemma.