# Showing a metric space to be incomplete [duplicate]

I'm supposed to show that the set of real numbers, $$R$$, is an incomplete metric space with metric $$d(x,y)=|\text{tan}^{-1}(x)-\text{tan}^{-1}(y)|$$.

My issue: What we usually do in such problems is that we find a cauchy sequence which doesn't converge in same metric space. Now since all of the terms of the sequence $$(x_n)_{n\geq1}$$ are supposed to come from real numbers, shouldn't the limit itself be in real numbers too? Also, we were taught a theorem in class "Every Cauchy sequence of real numbers converges" which means that the limit will exist too. Then why would it be incomplete? Thanks!

## marked as duplicate by José Carlos Santos real-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jan 28 at 9:52

Consider $$u_n=n$$, $$u_n$$ is a Cauchy sequence since $$lim_ntan^{-1}(n)=\pi/2$$ and does not have a limit since there does not exists a number $$x$$ with $$tan^{-1}(x)=\pi/2$$.