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.