Prove that A= { (x,1/x) ; x>0 } is closed: trying to show closure(A) is subset of A

pisrationalhahaha

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Prove that A= { (x,1/x) ; x>0 } is closed: trying to show closure(A) is subset of A

Prove that A= { (x,1/x) ; x>0 } is closed

I was trying to prove that Cl(A)A\displaystyle Cl(A)\subseteq A

So if I take (a,b)Cl(A)B((a,b),r)Aϕ\displaystyle (a,b)\in Cl(A)\Rightarrow B((a,b),r)\cap A\neq \phi for any r>0
Then I should prove that this couple belongs to A
Well here is the problem, I don't know how to continue
 
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I was trying to prove that Cl(A)A\displaystyle Cl(A)\subseteq A
So if I take (a,b)Cl(A)B((a,b),r)Aϕ\displaystyle (a,b)\in Cl(A)\Rightarrow B((a,b),r)\cap A\neq \phi for any r>0
Then I should prove that this couple belongs to A
Well here is the problem, I don't know how to continue
Lets use A  \displaystyle \overline{A\;} as the closure of A\displaystyle A . It actually easier to show the the completement Ac\displaystyle {A}^c is open.
If you are allowed to do that, note that if P:(p,q)A\displaystyle P: (p,q)\notin A then show the distance D(P,A)>0\displaystyle \mathscr{D}(P,A)>0
If not allowed, please tell us.
 
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Lets use A  \displaystyle \overline{A\;} as the closure of A\displaystyle A . It actually easier to show the the completement A  c\displaystyle \overline{A\;}^c is open.
If you are allowed to do that, note that if P:(p,q)A\displaystyle P: (p,q)\notin A then show the distance D(P,A)>0\displaystyle \mathscr{D}(P,A)>0
If not allowed, please tell us.

No not allowed
I don't even know such information yet
 
No not allowed. I don't even know such information yet
What a strange response? What kind of course are you taking?
In any course in which the concept of a closed set comes up then so does the idea of an open set.
Moreover, it is well known that the complement of an open set is a closed set & visa-versa.
 
What a strange response? What kind of course are you taking?
In any course in which the concept of a closed set comes up then so does the idea of an open set.
Moreover, it is well known that the complement of an open set is a closed set & visa-versa.
No I know this
But what you've said about the distance was unfamiliar
 
Okay I'll try to prove that the complement of A is open
So there would be two cases to consider
Case 1: y>1/x
Case 2: y<1/x (for any x>0)
If y>1/x:
Let (x,y)Ac\displaystyle \forall (x,y)\in A^{c}
B((x,y),r) for some r>0 be given
If (a,b)B((x,y),r)\displaystyle (a,b)\in B((x,y),r)
then
(xa)2+(yb)2<r2\displaystyle (x-a)^2+(y-b)^2< r^2
Knowing that (yb)2<(xa)2+(yb)2\displaystyle (y-b)^2<(x-a)^2+(y-b)^2.
then (yb)2<r2\displaystyle (y-b)^2<r^2 r<yb<r\displaystyle \Leftrightarrow -r<y-b<r
So now we have y-r<b<r+y
So if we choose r=y-1/x then we get that b>1/x, therefore (a,b) belongs to the complement of A
Is what I am doing right ?
 
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