Lemma 1

For each finite subset $I$ of $\mathbf{N}$, let $\varphi(I) = \sup_{J \subset I} ||p_J - e||$. Then for each subset $J$ of $I$ we have

The lemma is obvious if $J$ is empty; we shall prove it by induction on the number of elements in $J$. Let $J = K \cup \{j\}$, where $\{j\}$ is strictly larger than every $i \in K$. Then $p_J = p_K(e + a_j)$ and

so that

and by the inductive hypothesis and the definition of $\varphi(I)$, we have

which proves the lemma.