We can find M>0, such that each of the three sets have elements in (0,M], to simplify our notation, let X,Y,Z be the intersection of A,B,C with (0,M] respectively, and let x,y,z be the corresponding supremums.
Without loss of generality, we may assump x<=y<=z=M
case 1. x
case 2. x=y, then inf(Z\cap (x M]) = x, so we can find x1,y1,z1 from X,Y,Z respecively, so that they are close to x, with maximum difference less than x/9. Then it can be easily shown that x1,y1,z1 can be the lengths of the edges of a triangle.
Without loss of generality, we may assump x<=y<=z=M
case 1. x
case 2. x=y, then inf(Z\cap (x M]) = x, so we can find x1,y1,z1 from X,Y,Z respecively, so that they are close to x, with maximum difference less than x/9. Then it can be easily shown that x1,y1,z1 can be the lengths of the edges of a triangle.
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