I love maths, just not this question. need Help?

Question

Could some one tell me how to prove that for any A in the set (-1,0), then lim{n→∞} Aⁿ = 0, using just the definition of convergence

{ }brackets is subscript.

Answer 1

Let e > 0 be given.  We must prove there exists N > 0 such that if

n > N, then |0 - A^n| < e.  Notice that since A is in (-1,0), |A| < 1, so

ln(|A|) < ln(1) = 0.  Choose N > ln(e)/ln(|A|).  Now if

n > N > ln(e)/ln(|A|), then (since ln(|A|) < 0, we have n ln(|A|) < ln(e), or ln(|A|)^n < ln(e), so |A|^n < e.  That is  |0 - A^n| < e, so the limit is 0.