Let $M=(0,\infty)$ be supplied with the metric function $d(x,y)=|\arctan(x)-\arctan(y)|$ and let $\{n\}_{n=1}^\infty$ be a sequence of positive integers.
a) Is the sequence a Cauchy sequence in $(M,d)$?
b) Is the sequence a convergent sequence in $(M,d)$?
I am pretty sure it is Cauchy but not convergent. What I have so far is:
$|\arctan(n)-\arctan(m)|<\epsilon$ if $n,m$ are sufficiently large: $|\arctan(n)-\arctan(m)|≤|\arctan(n)-0|+|\arctan(m)-0|=\arctan(n)+\arctan(m)<\cdots$
This is where I'm stuck.
Well, since $$ \lim_{n\to+\infty} \arctan n = \frac{\pi}{2} $$
you get $$ d(m,n)\to \left\lvert\frac{\pi}{2}-\frac{\pi}{2}\right\rvert=0 $$