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**Anonymous1** · November 16th 2009, 05:21 PM

Consider the set $S$ of all real numbers, but with a new distance function defined:

$d(x,y) = |\frac{x}{1 + x} - \frac{y}{1 + y}|$

Add two new points, $+\infty$ and $-\infty$ to the set $S.$ call the resulting set $\bar{S} = S \cup \{+\infty,-\infty\}.$

Now extend $d$ to $\bar{S}$ by setting $d(x,+\infty) = |\frac{x}{1 + x}-1|, d(x,-\infty) = |\frac{x}{1 + x}-1|$ and $d(+\infty,-\infty) = 2.$

(a) Prove: a sequence of reals $x_n \in \bar{S}$ converges to $+\infty \iff x_n \rightarrow \infty.$

(b) Prove: The sequence $s_n \in \bar{S}$ converges $\iff$ it is $d-Cauchy.$

Any guidance would be greatly appreciated. Thanks!

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