Suppose we have two disjoint open balls in a euclidean space, $B(x,s)$ and $B(y,r)$. Then $d(x,y) \geq r+s$. I'd appreciate any help or hint. Thank you.
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2Hint: Draw two disjoint balls, for the picture disks. Join $x$ and $y$ by a straight line. That should guide one to the general argument. – André Nicolas Aug 04 '16 at 21:47
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I did that but and i assumed that d(x,y)<r+s. So i must find at least one point α that d(x,α)<r and d(y,α)<s for the contradiction but i get trouble to find one such point – Marios Gretsas Aug 04 '16 at 21:59
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Let $r+s=d(x,y)+b$. The point $p$ on the straight line joining the two centres, at distance $r-b/2$ from $x$ and therefore $s-b/2$ from $y$ is in both balls. Note that we will have to use some property of Euclidean spaces. For the result certainly does not hold in general metric spaces. – André Nicolas Aug 04 '16 at 22:11