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

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

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

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

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

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

Any guidance would be greatly appreciated. Thanks!