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## Main Question or Discussion Point

Hi.I have this trivial problem for a metric d(x,y) that d((x,y)≥0. My alternative proof is 2d(x,y)=√4d

^{2}(x,y)=√d^{2}(x,y)+d^{2}(y,x)+2d(x,y)d(y,x)=√(d(x,y)+d(y,x))^{2}≥d(x,x)=0 .Well it perhaps is a trivial proof but I did not know of this proof so I wanted to post it. Do you know other alternative proofs of this or other elementary or not so trivial problems in topology of metric spaces?The book I read had other proof for this problem.