What do the boundedness and monotonicity of a sequence have to do with limits?

George E. Hrabovsky, President, MAST

Where We Have Been

Las t time I wrote about the fundamental properties of sequences. This time I will link those ideas to our discussion of limits.

Session 5: What do the boundedness and monotonicity of a sequence have to do with limits?

Before we examine any resources, let's do a little bit of thinking. For a sequence to have a limit, it must arrive at some specific value eventually. For this to happen it seems likely that there must be some boundary on its upper and lower values. Thus, we can expect that any sequence possessing a limit is bounded.

Similarly it seems likely that the value of a sequence with a limit will approach that limit with each successive term.  It thus seems likely that such a sequence will be monotonic.

These are not theorems, we have not proved them.  They are assertions and should be seen more as suppositions than hard facts.

What is the limit of a sequence?

According to [1] the limit of a sequence is some value L that each successive term of the sequence approaches.

58. What does it mean for a sequence to approach the value L?

What does it mean for a sequence to approach the value L?

It means that gets ever closer to L.

59. How close does have to get to L?

How close does have to get to L?

The distance between and L must get within some positive real number, ε, from each other. This is expressed,

This is only true if the number of terms is sufficiently large. We can express this largeness by the symbol δ. So, we write,

This simply states that a sequence will tend to reach its limit (if such exists) as the number of terms of the sequence gets beyond a certain number.

References

[1] Donald W. Hight, "A Concept of Limits," Prentice-Hall, Inc.  (1966) (republished in 1977 by Dover Publications).