BOUNDED SEQUENCES
Analysis
 
4. DEFINITION OF A LOWER BOUND

A number is said to be a lower bound of a sequence if it is less than or equal to all the terms in the sequence.

9.- Note that -2 is a lower bound of this sequence and therefore, when represented on the Cartesian plane, the points in the sequence are above the line y=-2.

10.- Check that the same thing happens for any other lower bound k, i.e. the points in the sequence are above the line y=k. In other words, for any term n:

a

n

³

k

 or 

a

n

-

k

³

0

If a sequence has a lower bound it is said to be lower bounded.
   
5. DEFINITION OF AN UPPER BOUND

A number is said to be an upper bound of a sequence if it is greater than or equal to all the terms in the sequence.

11.- Note that 7 is an upper bound of this sequence and therefore, when represented on the Cartesian plane, the points in the sequence are below the line y=7.

12.- Check that the same thing happens for any other upper bound K, i.e. the points in the sequence are below the line y=K. In other words, for any term n:

a

n

£

K

 or 

a

n

-

K

£

0

If a sequence has an upper bound it is said to be upper bounded.
   
       
           
  Juan Madrigal Muga
 
© Spanish Ministry of Education and Education. Year 2002