Proof that a sequence that is equivalent to a cauchy sequence is also cauchy

[mrose14]

Need to prove that if a sequence is equivalent to a sequence that is assumed to be cauchy, then it is also cauchy.

 

[Steven G]

OK, so what have you tried? Where are you stuck? This is a help forum so you need to show us some work so we can figure out why what type of help you need.......................................[edited]

I would suggest that you figure out what is meant by equivalent in your post.

Please post back with the definition and your attempt at the proof.

 

[pka]

Need to prove that if a sequence is equivalent to a sequence that is assumed to be cauchy, then it is also cauchy.

Because Cauchy is a last name it is used with capital C. The statement that a sequence is Cauchy "informally" means that from some index (subscript) on "all of terms are close together". Please note that I have used quote makes "" and said these informal notions. But they are designed to help with the formal definitions.
Here is an example of what I mean.
The statement that the sequence \(x_n\) converges (has a limit) means that there is a number \(L\) such that almost all all but a finite collection of the \((x_n's)\) are close to \(L\). If \(x_n\to L\) means if \(\varepsilon>0\) then there exists a positive integer \(N\) such that for \(\forall j\ge N\) it is the case that \(\left|L-x_j\right|<\varepsilon\), That formal definition says that from \(N\) on the terms \(x_j\in\left(L-\varepsilon , L+\varepsilon\right)\).

Now it turns out because of the structure of the real numbers if a sequence is Cauchy it converges. That is known as completeness.