In high school, we have studied the monotonicity of sequence . For the convenience of application, the definition of monotonic sequence is briefly given below.

　　Definition: If the sequence satisfies the condition

,

Then the sequence is said to be monotonically increasing; if the sequence satisfies the condition

,

Then the sequence is said to be monotonically decreasing. Sequences that increase and decrease monotonically are collectively called monotonic sequences.

　　Criterion Ⅱ：A monotonic bounded sequence must have a limit.

　　We do not prove criterion II, but the following figure can us intuitively understand why a monotonically increasing and bounded sequence must have a limit . Because the sequence increases monotonically and cannot be greatera certain number , that is, for any given , there must be and number , so that when , there is always

,

Therefore, the limit of the sequence exists.

Monotonically increasing bounded sequence example [ animation ];

Example of monotonically decreasing bounded sequence [ animation ].

　　According to Theorem 1 in Properties of Convergent Sequences , a convergent sequence must be bounded. But bounded sequences do not necessarily converge. Criterion II states that if a sequence is not only bounded but also monotonic, then the sequence must converge.

[ Animation ]