Show that the enumeration diverges

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  • #1
PirateFan308
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Homework Statement


Let A = (0,1)[itex]\cap[/itex]Q
Let (Xn) be an enumeration of A. Show that (Xn) diverges

The Attempt at a Solution


I am not quite sure of what 'diverges' means. Does this simply mean that a tail of (Xn) diverges to +∞ or -∞? If this is what the definition is, I cannot see how (Xn) diverges. I would say instead that it converges to 1.
 
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  • #2
PirateFan308 said:

Homework Statement


Let A = (0,1)[itex]\cap[/itex]Q
Let (Xn) be an enumeration of A. Show that (Xn) diverges

The Attempt at a Solution


I am not quite sure of what 'diverges' means. Does this simply mean that a tail of (Xn) diverges to +∞ or -∞? If this is what the definition is, I cannot see how (Xn) diverges. I would say instead that it converges to 1.

The question claims (truly) that if you have *any* list containing all the rationals in (0,1) [which is possible because this is a countable set], the list is not a convergent sequence; in other words, x_n does not have a well-defined limit as n --> infinity. Of course, 0 < x_n < 1 for all n, so there is no question x_n going to +- infinity as you seem to think.

RGV
 

1. What does it mean for an enumeration to diverge?

An enumeration is said to diverge when the list or sequence being described continues infinitely without ever reaching a definite conclusion or limit. In other words, the terms in the enumeration keep increasing or changing without ever stabilizing or converging to a specific value.

2. How can I show that an enumeration diverges?

One way to show that an enumeration diverges is by demonstrating that the terms in the list or sequence increase without bound or become increasingly erratic. This can be done through mathematical proofs or by providing specific examples that illustrate the behavior of the enumeration.

3. What are some common methods for proving that an enumeration diverges?

Some common methods for proving that an enumeration diverges include using the limit comparison test, the ratio test, or the divergence test. These techniques involve analyzing the behavior of the terms in the enumeration and determining whether they approach a finite limit or continue to increase without bound.

4. Can an enumeration ever converge?

Yes, an enumeration can converge under certain conditions. For example, if the terms in the list or sequence decrease with each subsequent term and eventually approach a finite limit, then the enumeration can converge. However, if the terms increase or change without bound, the enumeration will diverge.

5. Why is it important to determine if an enumeration diverges or converges?

Understanding whether an enumeration diverges or converges is crucial in many mathematical and scientific fields. It can help in making predictions, analyzing patterns, and solving equations. In addition, knowing the behavior of an enumeration can provide insights into larger mathematical concepts and help in developing new theories or techniques.

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