Criterion 2 Monotone bounded sequence must have a limit.

The textbook of guideline 2 does not prove it, but gives the following geometric explanation:

From the number axis, the point xn corresponding to the monotone sequence can only move in one direction, so there are only two possible situations: either the point xn moves infinitely along the number axis (xn tends to be positive or negative infinity); Or the point xn infinitely approaches a fixed point a, that is, the sequence {xn} approaches a limit. But now assuming that the sequence is bounded and all the points xn of the bounded sequence fall within a certain interval [-M, M] on the number axis, then the above-mentioned first situation cannot happen. This means that this series tends to a limit, and the absolute value of this limit does not exceed m.

That is, the proof given in the geometric sense.

However, it should be noted that this is only limited to series, and it is wrong to say that monotone bounded functions must have limits.