I have two sequences:
By the Monotonic Sequence Theorem, I think that these are both convergent. The first sequence is bounded between 1 and 0 and is monotonic. Therefore it must converge right? But where? At 0?
The same seems to be true with the second sequence. The second sequence is also bounded between 1 and 0. It doesn't fit the definition of decreasing or increasing since it jumps up and down, so it must be monotonic. Yet it's gradually approaching a value of zero.
So it's not monotonic. It doesn't have a limit, so it is divergent correct?
But for the second sequence, it doesn't seem to fit the definition of monotonic either. Even though it is decreasing in general, it jumps up and down. So I can't say that .