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 $$tan^{-1}(x)=\pi/2$$.