Question on Brun's Work

Question

In Brun's Pure Sieve (using standard sieve theory notation), Brun showed that $$S(A,P,z)=A-\sum_{p\leq z}A_p+\sum_{p_1<p_2\leq z}A_{p_1p_2}-\dots\pm \sum_{p_1<p_2<p_3<\dots<p_{\pi(z)}\leq z}A_{p_1p_2\cdots p_{\pi(z)}}$$ If we call $$\Psi_i=\sum_{p_1<p_2<\dots<p_i\leq z}A_{p_1\cdots p_i} $$ Then Brun showed that if $i$ is even then $$\sum_{j=0}^{i-1}(-1)^j\Psi_j\leq S(A,P,z)\leq \sum_{j=0}^i (-1)^j\Psi_j$$ This means that the values of $\Psi_i$ must be larger than the absolute value of the previous partial sum up to $i-1$. This can mean that the values of $\Psi_i$ increase, or decrease or act semi-randomly. Does anyone know offhand if Brun did any work on showing that $\Psi_i$ is either increasing or decreasing ???