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Suppose that $(x_n)$ is a decreasing sequence of non-negative real numbers that converges to 0. Prove, using the definition, that the sequence $(y_n)$ where $y_n = x_1 - x_2 + x_3 - x_4 + \cdots + (-1)^{n{+1}}x_n$ is a Cauchy sequence.

I understand that a Cauchy sequence is a sequence that, for all $\epsilon < 0$ there exists a $p \in \mathbb{N}$ such that for all $m,n \in p$ $|x_m-x_n|< \epsilon$. However, I'm confused as to how to actually go about proving something is a Cauchy sequence and how in this case the sequence is built up from a different sequence. Any hints or suggestions would be appreciated, thank you!

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  • $\begingroup$ Start with writing an expression for $y_n - y_m$. $\endgroup$ – md2perpe Apr 25 at 18:18
  • $\begingroup$ After writing the expression $y_n-y_m$, try to bound its absolute value from above. Hint the monotonicity of $(x_n)$ is crucial here. $\endgroup$ – kneidell Apr 25 at 18:25
  • $\begingroup$ Would this be the correct expression for $y_n - y_m$ - $y_n - y_m = (x_1-x_2....(-1)^{n+1}x_n) -(x_1-x_2....(-1)^{m+1}x_m $ ? $\endgroup$ – Masha Apr 25 at 18:40
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Take $\varepsilon>0$. If $n$ is large enough, you have $x_N<\varepsilon$. Pick such a $N$. Now, if $m,n\geqslant N$, you want to prove that $\lvert y_m-y_n\rvert<\varepsilon$. This is trivial if $m=n$. If $m>n$, then$$y_m-y_n=\begin{cases}x_{n+1}-x_{n+2}+\cdots+\pm x_m&\text{ if }n+1\text{ is odd}\\-x_{n+1}+x_{n+1}-\cdots+\pm x_m&\text{ otherwise.}\end{cases}$$But, if $n+1$ is odd,$$y_m-y_n=x_{n+1}-(x_{n+2}-x_{n+3})-\cdots<x_{n+1}<\varepsilon$$and$$y_m-y_n=(x_{n+1}-x_{n+2})+(x_{n+3}-x_{n+4})+\cdots>0.$$Therefore, $\lvert y_m-y_n\rvert<\varepsilon$. The case in which $n+1$ is even is similar.

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  • $\begingroup$ Hi, I'm sorry I'm still a little confused - How did you get the expression for ym-yn if m>n? $\endgroup$ – Masha Apr 25 at 18:40
  • $\begingroup$ Suppose, for instance that $m=8$ and that $n=5$. Then\begin{align}y_8-y_5&=(x_1-x_2+x_3-x_4+x_5-x_6+x_7-x_8)-(x_1-x_2+x_3-x_4+x_5)\\&=-x_6+x_7-x_8.\end{align} $\endgroup$ – José Carlos Santos Apr 25 at 18:43
  • $\begingroup$ Thank you, I'm just still confused on a few parts. So, if $y_m - y_n_ = x_{n+1} -x_{n+2} + x_{n+3} + ... x_m} then how do we get that $y_m-y_n = x_{n+1}-(x_{n+1} - x_{n+3}) - .... What happened to the pluses and why did they turn into minuses? Thank you! $\endgroup$ – Masha Apr 29 at 21:18
  • $\begingroup$ What you wrote is unreadable and I turned no $+$ into a $-$ or a $-$ into a $+$. $\endgroup$ – José Carlos Santos Apr 29 at 21:25
  • $\begingroup$ My apologies, I'm still learning how to use the site. To rephrase, my question is for $n+1$ is odd, how did you get that $𝑦_𝑚−𝑦_𝑛=𝑥_{𝑛+1}−(𝑥_{𝑛+2}−𝑥_{𝑛+3})−⋯<𝑥_{𝑛+1}<𝜀$ if, from above that, we have that $𝑦_𝑚−𝑦_𝑛=𝑥_{𝑛+1}−𝑥_{𝑛+2}+𝑥_{𝑛+3}−x_m$ if $n+1$ is odd $\endgroup$ – Masha Apr 29 at 21:28

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