Is xn+yn a Cauchy sequence if xn and yn are Cauchy sequences?

In summary, to prove that the sequence xn+yn is Cauchy without using the Cauchy Criterion or the Algebraic Limit Theorem, we can use a direct argument with the triangle inequality and the properties of Cauchy sequences.
  • #1
kathrynag
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Homework Statement



Let xn and yn be Cauchy sequences.
Give a direct argument that xn+yn is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem.

Homework Equations





The Attempt at a Solution


given epsilon>0 there exists an N in the natural numbers such that whenever m,n>N, it follows that:
[tex]\left|xn-xm\right|<epsilon[/tex] and [tex]\left|yn-ym\right|[/tex]<epsilon
I'm not sure where to go next.
 
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  • #2
To show that [tex] x_n + y_n [/tex] is Cauchy, you want to prove that for any [tex] \epsilon > 0 [/tex] there is a natural number [tex] N [/tex] such that if [tex] n, m > N [/tex] then [tex] |(x_n + y_n) - (x_m + y_m)| < \epsilon [/tex], right?

Note that [tex] |(x_n + y_n) - (x_m + y_m)| = |(x_n - x_m) + (y_n - y_m)| [/tex]. Now use the triangle inequality and remember that both sequences [tex] x_n, y_n [/tex] are Cauchy.
 

Related to Is xn+yn a Cauchy sequence if xn and yn are Cauchy sequences?

What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers where the terms get closer and closer together as the sequence progresses. This means that for any given margin of error, there exists a point in the sequence where all subsequent terms fall within that margin.

How do you prove that Xn + yn is a Cauchy sequence?

To prove that Xn + yn is a Cauchy sequence, you would need to show that for any given margin of error, there exists a point in the sequence where the sum of the terms falls within that margin. This can be done by using the definition of a Cauchy sequence and algebraic manipulations.

What is the importance of Xn + yn being a Cauchy sequence?

Xn + yn being a Cauchy sequence is important because it shows that the sequence is convergent, meaning it approaches a single limit as n approaches infinity. This is useful in many areas of mathematics, particularly in calculus and analysis.

Are all Cauchy sequences convergent?

No, not all Cauchy sequences are convergent. While all convergent sequences are Cauchy, the converse is not always true. This means that some Cauchy sequences may not have a limit, and therefore are not convergent.

Can Xn + yn be a Cauchy sequence if Xn and yn are not Cauchy sequences on their own?

Yes, it is possible for Xn + yn to be a Cauchy sequence even if Xn and yn are not Cauchy sequences on their own. This is because the sum of two non-convergent sequences can still be a convergent sequence if the terms of the two sequences cancel each other out as n approaches infinity.

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