Let ($X,d$) be a metric space and let $<x_n>$ and $<y_n>$ be arbitrary Cauchy sequences in $X$.Which of the following statement is true?
a.The sequence {$d(x_n,y_n)$} converges as $n\rightarrow\infty$.
b.The sequence {$d(x_n,y_n)$} converges as $n\rightarrow\infty$ only if $X$ is complete.
Solution: the proof for part (a) is same as amsmath's answer
What is the counterexample for (b) part?
On
Take
$$X=\Bbb Q$$ $$d=| \;\; |$$
$$x_n=1+\frac 11 +\frac{1}{2!}+...+\frac{1}{n!}$$
$$y_n=2x_n$$
$$\lim_{n\to+\infty}|y_n-x_n|=e\notin \Bbb Q$$
$(x_n)$ and $(y_n)$ are Cauchy in $\Bbb Q$ but the differene $(y_n-x_n)$ is not convergent in $\Bbb Q$.
On
For an extremely simple counterexample, let $(X,d)=( (0,1), |\cdot|)$ Let $x_n=\frac{1}{n+1}$ and $y_n=1-\frac{1}{n+1}$. Then $(x_n)$ and $(y_n)$ are Cauchy sequences in $(0,1)$ that do not converge in $(0,1)$, meaning that $(0,1)$ is not complete. But:
$$|x_n-y_n|=\left|1-\frac{1}{n+1}-\frac{1}{n+1}\right|=\left|1-\frac{2}{n+1}\right|\rightarrow 1\ as\ n\rightarrow\infty$$
Get $\mathbb{Q}$ with the Euclidean distance. Then if $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ are arbitrary Cauchy sequences in $\mathbb{Q}$, they are also Cauchy sequences in $\mathbb{R}$, so by the completeness of $\mathbb{R}$ they have limit in $\mathbb{R}$, say to $x\in\mathbb{R}$ and to $y\in\mathbb{R}$ respectively. So: $$d(x_n,y_n)=|x_n-y_n|\rightarrow|x-y|, n\rightarrow\infty.$$ On the other hand, $\mathbb{Q}$ with the Euclidean distance is not a complete metric space, because for example the truncated decimal expansion of $\sqrt2$ is a Cauchy sequence in $\mathbb{Q}$ that doesn't admit limit in $\mathbb{Q}$.