3.1. Sequences
Example 3.1.3(c):
{ n / 2n } = { 0, 1/2, 1/2, 3/8, 1/4, 5/32, ...}. It is not clear, but it seems as if the terms get smaller and smaller. Indeed this is the case, and we will prove it:
First, we can use induction to show that
n2 2nfor n > 3. But then we have that
n2 / 2n 1or equivalently
n / 2n 1/nfor n > 3. But now you should be able to finish the proof yourself. As a hint, for a given , choose
N = max{3, 1/}