The identity of \(\ell _2^{\, n+1}\) factors as \(\mathrm{id}_{\ell _2^{\, n+1}} = \mathrm{proj}_\infty \circ I \circ \mathrm{emb}_1\) with \(\| \mathrm{emb}_1\| , \| \mathrm{proj}_\infty \| \le \sqrt{n+1}\). By the ideal property of the Hilbert numbers and their normalisation \(h_n(\mathrm{id}_{\ell _2^{\, n+1}}) \ge 1\),

Let \(A \in \mathcal{L}(\ell _2, \ell _1)\) and \(B \in \mathcal{L}(\ell _\infty , \ell _2)\) be non-zero; we must bound \(a_n(B I A) \le \| A\| \, \| B\| /(n+1)\). Factorising \(I = J_2 J_1\) with the norm-one inclusions \(J_1 \colon \ell _1 \to \ell _2\) and \(J_2 \colon \ell _2 \to \ell _\infty \), we obtain \(T := B I A = T_2 T_1\) with \(T_1 := J_1 A\) and \(T_2 := B J_2\) on \(\ell _2\). Applying Lemma 8.8 to \(B\) and to the rows of \(A\),

which by Lemma 8.9 gives the two hypotheses of Theorem 8.10 along the singular vectors of \(T\). That theorem yields \((n+1)\, a_n(T) \le \| A\| \, \| B\| \) provided \(T\) is compact.

For compactness, replace \(A\) by its coordinate truncation \(A_m\), of rank \(\le m\) and with \(\| A_m\| \le \| A\| \); then \(B I A_m\) has finite rank, hence is compact (Theorem 6.3), and the estimate above applies to it. Since \(\sum _j \| \mathrm{row}_j(A)\| ^2 {\lt} \infty \) we have \(\| \mathrm{row}_j(A)\| \to 0\), so \(\| I(A - A_m)\| \to 0\), and subadditivity of the approximation numbers,

lets \(m \to \infty \). Dividing by \(\| A\| \, \| B\| \) and taking the supremum over all admissible \(A, B\) gives \(h_n(I) \le (n+1)^{-1}\).

Combine Proposition 8.12 and Theorem 8.13.