Note: This is a homework problem.

Suppose that { } is a bounded sequence of reals such that for any n, .

Show that is an increasing sequence and, converges.

I've already done the first part, that is an increasing sequence. For the second part, my attempt so far has been that looking at , there are two possibilities:

Case 1: It's possible that . In that case, if for some integer k, , then we can use that to show that , and so , establishing by induction that { } is also an increasing bounded sequence and then convergent.

Case 2: It's also possible that . This is where I'm having trouble. The only way I can see to continue with this is to show that the sequence { } is decreasing in this case. After several failed attempts, I'm starting to become skeptical that it actually is decreasing. Is this not going to work, and should I abandon this attempt? If so, does anyone have a suggestion for what direction to go in?