Lemma 1

For each finite subset $J$ of $I$, let $\varphi(J) = \sup_{L \subset J} (p_L - 1)$. Then for each subset $L$ of $J$ we have

This is clear if $L$ is empty. We proceed by induction on card $(L)$. Let $L = K \cup \{ \lambda \}$, where $\lambda \notin K$; then $p_L = p_K (1 + a_\lambda)$ and $s_L = s_K + a_\lambda$, so that $p_L - 1 - s_L = (p_K - 1 - s_K) + (p_K - 1)a_\lambda$; hence, by the inductive hypothesis and the definition of $\varphi(J)$, we have

which proves the lemma.