For large enough $i$ we can choose $\chi^{i}$ with the following two properties.

• •

  $d(\chi^{i}(z),\chi(z))\leq\rho k(p)^{-1/2}/100$

• •

  $\|g_{i}-g_{0}\|_{\tilde{U}},\|J_{i}-J_{0}\|_{\tilde{U}}\leq\tilde{\psi}$.