1. The problem statement, all variables and given/known data Let e be the number close to sqrt(a) by Newtons Method (That is picking a number, diving a by it, and taking their average, divide a by average, get a number, find their average, so on). Using |e<sqrt(a)+e| prove that if |a/e-e|<1/10 then |sqrt(a)-e|<1/10 Note that e is using the Newtons method a few times, not necessarily infinity, for any number of times. Also this is about positive integers, and 0 only, root and a. 2. Relevant equations 3. The attempt at a solution So were trying to prove the second one smaller then first (I think), that is: |sqrt(a)-e|<|a/e-e| sqrt(a)-e<a/e-e (since both are positive, as using the given inequality subtract e from both sides) so sqrt(a)<a/e e*sqrt(a)<a, but e is not necessarily smaller then sqrt a, what am I missing?