# Show that a sequence is a Cauchy sequence

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!

• Start with writing an expression for $y_n - y_m$. – md2perpe Apr 25 at 18:18
• 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. – kneidell Apr 25 at 18:25
• Would this be the correct expression for $y_n - y_m$ - $y_n - y_m = (x_1-x_2....(-1)^{n+1}x_n) -(x_1-x_2....(-1)^{m+1}x_m$ ? – Masha Apr 25 at 18:40

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,$$y_m-y_n=x_{n+1}-(x_{n+2}-x_{n+3})-\cdotsand$$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.

• Hi, I'm sorry I'm still a little confused - How did you get the expression for ym-yn if m>n? – Masha Apr 25 at 18:40
• 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} – José Carlos Santos Apr 25 at 18:43
• 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! – Masha Apr 29 at 21:18
• What you wrote is unreadable and I turned no $+$ into a $-$ or a $-$ into a $+$. – José Carlos Santos Apr 29 at 21:25
• 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 – Masha Apr 29 at 21:28