Lemma 1

For every finite subset $I$ of $\mathbf{N}$, let $\varphi(I) = \sup_{J \subset I} \|p_J - \mathbf{e}\|.$ For every subset $J$ of $I$, one has

The lemma is evident if $J$ is empty; let us prove it by induction on the number of elements of $J$. Let $J = K \cup \{j\}$, where $j$ is strictly upper to all the elements of $K$; then $p_J = p_K(\mathbf{e} + a_j)$ and $s_J = s_K + a_j$, whence and, by virtue of the induction hypothesis and the definition of $\varphi(I)$ which proves the lemma.