Is Every Sequence with a Cauchy Subsequence Also Cauchy?

[ozkan12]

Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?

 

[Sudharaka]

Let $\left\{{x}_{n}\right\}$ be a sequence...İf $\left\{{x}_{2n}\right\}$ is caucy sequence, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence ?

Hi ozkan12,

I don't think so. For example consider the alternating sequence, $\{x_n\}_{n=0}^\infty=\{(-1)^n\}_{n=0}^\infty$. Now $\{x_{2n}\}_{n=0}^\infty$ is a constant sequence which is Cauchy but $x_n$ is not.

 

[ozkan12]

Hi Sudharaka,

İn some fixed point theorem, to prove that $\left\{{x}_{n}\right\}$ is cauchy sequence, author show that $\left\{{x}_{2n}\right\}$ is cauchy sequence...And in fixed point theorems, we use iteration sequence such that ${x}_{n}=f{x}_{n-1}$...İf we construct $\left\{{x}_{n}\right\}$ in this way, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence by proving that $\left\{{x}_{2n}\right\}$ is cauchy sequence ?

 

[Sudharaka]

Hi Sudharaka,

İn some fixed point theorem, to prove that $\left\{{x}_{n}\right\}$ is cauchy sequence, author show that $\left\{{x}_{2n}\right\}$ is cauchy sequence...And in fixed point theorems, we use iteration sequence such that ${x}_{n}=f{x}_{n-1}$...İf we construct $\left\{{x}_{n}\right\}$ in this way, can we say that $\left\{{x}_{n}\right\}$ is cauchy sequence by proving that $\left\{{x}_{2n}\right\}$ is cauchy sequence ?

Could you please tell me which book you are referring to so that I can have a look?

 

[ozkan12]

Sudharaka, I learned this information, Thank you for your attention, best wishes...:)

 

[Sudharaka]

Sudharaka, I learned this information, Thank you for your attention, best wishes...:)

Sure, I have marked the thread as SOLVED. :)