1. The problem statement, all variables and given/known data Hi Guys, This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help. "E1. Starting with a point S (a, b) of the plane with 0 < b < a, we generate a sequence of points (xn, yn) according to the rule x0 = a, y0 = b, xn+1 =(xn + yn) / 2 and yn+1 = (2xnyn) / (xn + yn). Here it is easy to find an invariant. From (xn+1yn+1) = xnyn, for all n we deduce xnyn = ab for all n. This is the invariant we are looking for. Initially, we have y0 < x0. This relation also remains invariant. Indeed, suppose yn < xn for some n. Then xn+1 is the midpoint of the segment with endpoints yn, xn. Moreover, yn+1 < xn+1 since the harmonic mean is strictly less than the arithmetic mean. Thus, 0 < xn+1 − yn+1 = [(xn − yn) / (xn + yn)] * [(xn − yn) / 2] < (xn − yn) / 2 for all n. So we have limxn = lim yn = x with x2 = ab or x = √ab. Here the invariant helped us very much, but its recognition was not yet the solution, although the completion of the solution was trivial." 3. The attempt at a solution I cannot figure out the bit in bold at all. It says lim xn = lim yn, but where does this come from? From a cursory look at the definition of a limt, is it simply since |xn+1 - yn+1| < (xn-yn)/2, we find that for all N>n, that xn+1 and yn+1 are limits of each other?