1. The problem statement, all variables and given/known data given |a| < 1, show that the limit of |a|^n goes to 0 as n goes to infinity. 2. Relevant equations 3. The attempt at a solution let |a|<1 and n>0 (n is a natural number, a is a real number) then |a^n| < 1^n then |a|^n < 1 then 1/n * |a|^n <= |a|^n < 1 (and we know from earlier that 1/n * |a|^n goes to zero) so 0 <=|a|^n < 1 meaning |a|^n is bounded below. now note that |a| > |a|^2 > |a|^3 > ... |a|^n > |a|^(n+1) > ... since a < 1 Since a^n is progressively smaller and bounded below by zero, we know that a^n does converge by the convergent monotone sequence thm and since there are infinite items in this sequence, its limit cannot be in the set of a^n. therefore let L be the limit, |a^n - L| < e, since L is smaller than a^n; a^n - L < e. **I tried using the squeeze theorem but could not find a sequence that I new to be greater than a^n and goes to zero.** And I don't know what I am missing in order to be able to claim that L is zero.