Monotonic Sequence Theorem Question

[DivGradCurl]

Suppose you know that $$\left{ a_n \right}$$ is a decreasing sequence and all its
terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can
you say about the value of the limit?

My Answer:

This sequence has a limit because it is both bounded and monotonic, as it is stated in the
Monotonic Sequence Theorem. The minimum value of this particular limit must be 5.

My question:

Is that thoroughly answered? Did I miss anything?

 

[arunma]

Suppose you know that $$\left{ a_n \right}$$ is a decreasing sequence and all its
terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can
you say about the value of the limit?

My Answer:

This sequence has a limit because it is both bounded and monotonic, as it is stated in the
Monotonic Sequence Theorem. The minimum value of this particular limit must be 5.

My question:

Is that thoroughly answered? Did I miss anything?

Well, you can say that the limit is greater than or equal to five. As you know, a certain theorem says that any bounded, monotonic sequence has a limit. But if the problem is stated as you have written it, I don't think you can say that the limit must be 5.

All of the terms lie between 5 and 8. But do you know if the infimum of the sequence is 5? If the infimum is 5, then yes, the limit is also 5. But if 5 is simply a lower bound (and not necessarily the infimum), then you can't say that the limit is 5.

 

[DivGradCurl]

I think get it. The limit is not necessarily 5.

Thanks.