Prove that A

Prove that A - (x, y) there exists an r>0 such...

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Prove that A={(x,y)} 2 |x>0} is an open set. SOLUTION Intuitively, this set is open, because no points on the “boundary”, x=0, are contained in the set. Such an argument will often suffice after one becomes accustomed to the concept of openness. At first, however, we should give details. To prove that A is open, we show that for every point
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Unformatted text preview: (x, y) there exists an r>0 such that D r (x,y). If (x,y)A then x>0. Choose r=x. If (x1, y1) D r (x,y), we have: |x1-x|=sqrt((x1-x)^2)sqrt((x1-x)^2+(y1-y)^2)<r=x, and so x1-x<x and x-x1<x. The latter inequality implies x1>0, that is, (x1,y1)A. Hence D r (x,y) A and therefore, A is open....
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