In high school, we have introduced the boundedness of sequence . Next, we will further study the boundedness of sequence and the boundedness of convergent sequence.

　　Define：the logarithmic sequence . If there is a positive number , then for all natural numbers , there is always

,

Then the sequence is said to be bounded, otherwise, it is said to be unbounded.

　　For example, the sequence

is bounded because can be taken such that

for all positive integers This holds true  .

　　The sequence is unbounded because when increases infinitely, can exceed any positive number.

　　Geometrically, if the sequence is bounded, then there exists such that the points on the number axis corresponding to the bounded sequence all fall on the closed interval .

　　Theorem 1：The convergent sequence must be bounded.

　　Proof：Suppose is defined by the limit . If is taken , then exists , so that when , there is always

,

Using the basic absolute value inequality , we have

.

If you remember

,

Then for all positive integers , we have

,

Therefore, the sequence is bounded.

　　Corollary 1：Unbounded sequence must diverge.

　　In fact, the above inference is the converse proposition of the theorem .