1. The problem statement, all variables and given/known data Prove that the real numbers a and b are equal if and only if for each positive real e, the absolute value of a-b satisfies abs(a-b)<e 2. Relevant equations The main one I am thinking about is the fact that if a<=b and b<=a then a=b, also the whole sigma thing might mean the archimedean property might come into effect (if r and s are positive ration numbers then there exists a positive integer N such that rN<s). 3. The attempt at a solution I am tried a proof by contradiction.... but generally it falls apart or requires too much non-rigorous work :-/ Any ideas/hints/help? Thanks!