I understand how to do the proof for an increasing sequence and I could figure i
ID: 1892476 • Letter: I
Question
I understand how to do the proof for an increasing sequence and I could figure it out for a decreasing sequence but I am lost on how to prove this for an eventually decreasing sequence. I am not sure how the epsilon fits in and would you set L to equal the sup S or L = inf S when S= {a_n | n E N}Much help would be helpful.
Prove that for an eventually decreasing sequence {a_n}, there are two possibilities, either
- {a_n} is bounded below by M, in which case there exists L greater than or equal to M
such that lim n-> infinty a_n= L.
- {a_n} is unbounded, in which case lim n-> infinty an= negative infinty
Explanation / Answer
Let an be a sequence of complex numbers that converge to zero. Can we always find sn?{-1,1} such that ?8n=1snan converges? If the an are real numbers, we can find such a sequence sn. If the partial sum of the first N terms is positive we make sure the following terms are negative until the sum becomes less than zero. Then we switch to making the terms positive until the partial sum becomes greater than zero, and so on. It is easy to see that the partial sums will either tend monotonically towards zero, or oscillate around zero with decreasing amplitude.