1. The problem statement, all variables and given/known data Let a_n = (1/2)[(1/a_n)+1] and a_1=1, does this sequence converge? 2. Relevant equations A sequence in R^n is convergernt if and only if it's cauchy. A sequence in R^n is called a cauchy sequence if x_k - x_j ->0 as k, j-> infinity. 3. The attempt at a solution I am confused about what the solution says below: a_2 = 1, so all points a_i =1. therefore the sequence is equal to 1,1,1,1 ... (so far I am ok) this sequence converges to 1 by the cauchy condition: a_n+1 - a_n = 0 which goes to 0 as n goes to infinity. since limit points are unique, the sequence converges to 1. But as the definition of cauchy sequence says, it tends to 0 but should not equal to 0 as k,j goes to infinity. but here, all the elements are equal to 1, so x_k - x_j = 1 for all k,j. That's not the condition of limit, is it? cuz limit shouldn't be "equal", limit just can be tended to, right?