Proving Cauchy Sequence of |an-bn|

Thus (cn) is Cauchy.In summary, if (an) and (bn) are Cauchy sequences, then (cn) = |an - bn| is also a Cauchy sequence. This is because of the triangle inequality, which states that the absolute value of the difference between two numbers is less than or equal to the absolute value of their sum. By using this inequality, we can show that the sequence (cn) is also Cauchy, since the difference between the terms of (cn) is always smaller than a given value e.
  • #1
rbzima
84
0
I'm basically trying to show that if (an) and (bn) are Cauchy sequences, then (cn) = |an - bn| is also a Cauchy sequence.

I know that the triangle inequality is going to be used at one point or another, but I suppose I'm a little confused because:

(an) is Cauchy implies |an - am| < e
(bn) is Cauchy implies |bn - bm| < e

I think at some point my e's are going to be changed to e/2, which is totally legitimate because e is arbitrary anyway.
 
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  • #2
The key to this is to show that ||X|-|Y||<=|X-Y|

Then ||an-bn|-|am-bm||<=|(an-am)-(bn-bm)|<=|an-am|+|bn-bm|
 

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers in which the terms become arbitrarily close to each other as the sequence progresses. In other words, for any small positive number, there exists a point in the sequence after which all the terms are within that distance from each other.

2. Why is proving the Cauchy sequence of |an-bn| important?

Proving the Cauchy sequence of |an-bn| is important because it is a fundamental concept in real analysis and is used to prove the convergence of a sequence. It is also a key step in proving the completeness of the real numbers.

3. How do you prove the Cauchy sequence of |an-bn|?

To prove the Cauchy sequence of |an-bn|, you must show that for any small positive number, there exists a point in the sequence after which the absolute difference between any two terms is less than that number. This can be done using the triangle inequality and the properties of limits.

4. What are some common methods used to prove the Cauchy sequence of |an-bn|?

Some common methods used to prove the Cauchy sequence of |an-bn| include the use of the definition of a Cauchy sequence, the Cauchy criterion, the squeeze theorem, and the properties of limits.

5. Can a Cauchy sequence fail to converge?

Yes, a Cauchy sequence can fail to converge if it is not a convergent sequence. This can happen if the sequence is not defined for all real numbers or if there is a gap in the sequence. In such cases, the sequence is said to be divergent.

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