Is {yn} a Cauchy Sequence if {xn} Is?

[bloynoys]

Homework Statement

Prove the following assertion: Suppose {xn} and {yn} are Cauchy sequences of real number. If {xn} is a cauchy sequence and for every η>0 there exists a pos. int. N such that for every n>N so that abs(xn-yn)<η then {yn} is a Cauchy sequence.

Homework Equations

None

The Attempt at a Solution

We will prove that {yn} is a cauchy sequence by showing that for every ε>0 there exists a pos. int. N so that both n and m >N so that abs(yn-ym)<ε.

Consider ε>0 arbitrary.
Since {xn} is a cauchy sequence and by hypothesis for every η>0 there exists an N so that every n>N abs(xn-yn)<η. Choose such an N.
Consider n,m>N arbitrary.

Then I know I need to get from abs(yn-ym) to ε but unsure how to use what I have to get there correctly and if I have the rest of the proof right.

 

[sunjin09]

Homework Statement

Prove the following assertion: Suppose {xn} and {yn} are Cauchy sequences of real number. If {xn} is a cauchy sequence and for every η>0 there exists a pos. int. N such that for every n>N so that abs(xn-yn)<η then {yn} is a Cauchy sequence.

Homework Equations

None

The Attempt at a Solution

We will prove that {yn} is a cauchy sequence by showing that for every ε>0 there exists a pos. int. N so that both n and m >N so that abs(yn-ym)<ε.

Consider ε>0 arbitrary.
Since {xn} is a cauchy sequence and by hypothesis for every η>0 there exists an N so that every n>N abs(xn-yn)<η. Choose such an N.
Consider n,m>N arbitrary.

Then I know I need to get from abs(yn-ym) to ε but unsure how to use what I have to get there correctly and if I have the rest of the proof right.

Use the Cauchy sequence {xn} as your bridge, think about triangle inequality, also note that η may be set according to ε