To prove that a sequence is monotonic and bounded, mathematical induction and mathematical analysis techniques are usually needed. Here are some steps that might be useful:

First of all, we must determine the monotonicity of the sequence, that is, whether the sequence is monotonically increasing or decreasing. If the sequence is monotonically increasing, then for any ninN^*, there is angeqa{n- 1}. If the sequence is monotonically decreasing, then for any ninN^*, there is an {n+ 1}leqa{n}.

Next, you need to determine whether the sequence is bounded. If the sequence is bounded, there is a limit L, which makes ε > for any positive number; 0, with a positive integer n, so that when n >; When n, |a{n}-L|

Therefore, if a sequence is monotonically increasing or decreasing and bounded, it is monotonically bounded.