▁is ▁locally ▁Li ps ch itz ▁continuous ▁( for ▁instance ▁if ▁ ▁is ▁twice ▁differenti able ). ▁That ▁is , ▁it ▁is ▁assumed ▁that ▁for ▁any ▁open ▁subset ▁ ▁there ▁exists ▁a ▁constant ▁ ▁such ▁that ▁for ▁any ▁▁ ▁holds . ▁The ▁norm ▁on ▁the ▁left ▁is ▁some ▁operator ▁norm ▁that ▁is ▁compatible ▁with ▁the ▁vector ▁norm ▁on ▁the ▁right . ▁This ▁inequality ▁can ▁be ▁re written ▁to ▁only ▁use ▁the ▁vector ▁norm . ▁Then ▁for ▁any ▁vector ▁ ▁the ▁inequality ▁▁▁▁ ▁must ▁hold . ▁ ▁Now ▁choose ▁any ▁initial ▁point ▁. ▁Assume ▁that ▁ ▁is ▁invert ible ▁and ▁construct ▁the ▁Newton ▁step ▁▁ ▁The ▁next ▁assumption ▁is ▁that ▁not ▁only ▁the ▁next ▁point ▁ ▁but ▁the ▁entire ▁ball ▁ ▁is ▁contained ▁inside ▁the ▁set ▁. ▁Let ▁ ▁be ▁the ▁Li ps ch itz ▁constant ▁for ▁the ▁Jacob ian ▁over ▁this ▁ball . ▁ ▁As ▁a ▁last ▁prepar ation , ▁construct ▁recurs ively , ▁as ▁long ▁as ▁it ▁is ▁possible , ▁the ▁sequences ▁, ▁, ▁ ▁according ▁to ▁ ▁Stat ement ▁ ▁Now ▁if ▁ ▁then ▁ ▁a ▁solution ▁ ▁of ▁ ▁exists ▁inside ▁the ▁closed ▁ball ▁ ▁and ▁ ▁the ▁Newton ▁iteration ▁starting ▁in ▁ ▁converges ▁to ▁ ▁with ▁at ▁least ▁linear ▁order ▁of ▁convergence . ▁ ▁A ▁statement ▁that ▁is ▁more ▁precise ▁but ▁slightly ▁more ▁difficult ▁to ▁prove ▁uses ▁the ▁roots ▁ ▁of ▁the ▁quadratic ▁polynomial ▁, ▁ ▁and ▁their ▁ratio ▁▁ ▁Then ▁a ▁solution ▁ ▁exists ▁inside ▁the ▁closed ▁ball ▁ ▁it ▁is ▁unique ▁inside ▁the ▁bigger ▁ball ▁ ▁and