I would like to show that the sequence diverges. Here is my attempt.
The sequence converges to some element iff for all such that . My aim is to show that there is an epsilon for which no positive integer N satisfies this requirement.
Choose . Then there exists an N such that if n > N then .
Now let k be a positive odd integer such that k > N, then
and by a similar argument we can show that if we take k to be some even integer such that k > N then
But we can always find an even AND odd k such that the following inequalities are false for any real number L.
However there's something in the back of my mind which I'm not happy with. I don't like how I finished the proof, it doesn't seem quite sound to me. I was wondering if someone could point me in the right direction to finish it off or let me know where I've gone wrong and perhaps suggest a better approach.
Thanks in advance