 completes the proof.  Pathwise upper bounds on $|X(N^{0})|_{r,s}$ for $s \le N_{H_{r}}$ and $\Delta_{r}>0/b^{0}$ {#sec0} ======================================================  In the previous section, we analyzed the behavior of $|X(N^{0})|_{r,s} - |x|_{L}$ for $s