Show that a set is dense
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We want to show that the set $Ssubset mathbb{R}$ defined in the following way is dense: $S={x: exists c>0 text{ such that} |x-j/2^k| geq c/2^k, forall jinmathbb{Z},forall kin mathbb{N}_0 }$.
I tried to use the definition of a dense set, so for every $xin mathbb{R}$ the ball $B(x,varepsilon) cap Sneq emptyset$ for every $varepsilon >0$, but did not get anywhere.
real-analysis calculus
asked Dec 3 '18 at 1:26
elrondelrond
1336
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add a comment |
$begingroup$
We want to show that the set $Ssubset mathbb{R}$ defined in the following way is dense: $S={x: exists c>0 text{ such that} |x-j/2^k| geq c/2^k, forall jinmathbb{Z},forall kin mathbb{N}_0 }$.
I tried to use the definition of a dense set, so for every $xin mathbb{R}$ the ball $B(x,varepsilon) cap Sneq emptyset$ for every $varepsilon >0$, but did not get anywhere.
real-analysis calculus
asked Dec 3 '18 at 1:26
elrondelrond
1336
$endgroup$
$begingroup$
Not sure if it matters, but is $c$ a positive real number or a nonzero natural?
$endgroup$
– jgon
Dec 3 '18 at 2:35
$begingroup$
We have $c$ be a positive real, I edited the question.
$endgroup$
– elrond
Dec 3 '18 at 2:38
add a comment |
$begingroup$
We want to show that the set $Ssubset mathbb{R}$ defined in the following way is dense: $S={x: exists c>0 text{ such that} |x-j/2^k| geq c/2^k, forall jinmathbb{Z},forall kin mathbb{N}_0 }$.
I tried to use the definition of a dense set, so for every $xin mathbb{R}$ the ball $B(x,varepsilon) cap Sneq emptyset$ for every $varepsilon >0$, but did not get anywhere.
real-analysis calculus
asked Dec 3 '18 at 1:26
elrondelrond
1336
$endgroup$
We want to show that the set $Ssubset mathbb{R}$ defined in the following way is dense: $S={x: exists c>0 text{ such that} |x-j/2^k| geq c/2^k, forall jinmathbb{Z},forall kin mathbb{N}_0 }$.
I tried to use the definition of a dense set, so for every $xin mathbb{R}$ the ball $B(x,varepsilon) cap Sneq emptyset$ for every $varepsilon >0$, but did not get anywhere.
real-analysis calculus
real-analysis calculus
asked Dec 3 '18 at 1:26
elrondelrond
1336
asked Dec 3 '18 at 1:26
elrondelrond
1336
edited Dec 3 '18 at 2:38
elrond
asked Dec 3 '18 at 1:26
elrondelrond
1336
asked Dec 3 '18 at 1:26
elrondelrond
1336
asked Dec 3 '18 at 1:26
elrondelrond
1336
1336
$begingroup$
Not sure if it matters, but is $c$ a positive real number or a nonzero natural?
$endgroup$
– jgon
Dec 3 '18 at 2:35
$begingroup$
We have $c$ be a positive real, I edited the question.
$endgroup$
– elrond
Dec 3 '18 at 2:38
add a comment |
$begingroup$
Not sure if it matters, but is $c$ a positive real number or a nonzero natural?
$endgroup$
– jgon
Dec 3 '18 at 2:35
$begingroup$
We have $c$ be a positive real, I edited the question.
$endgroup$
– elrond
Dec 3 '18 at 2:38
$begingroup$
Not sure if it matters, but is $c$ a positive real number or a nonzero natural?
$endgroup$
– jgon
Dec 3 '18 at 2:35
$begingroup$
Not sure if it matters, but is $c$ a positive real number or a nonzero natural?
$endgroup$
– jgon
Dec 3 '18 at 2:35
$begingroup$
We have $c$ be a positive real, I edited the question.
$endgroup$
– elrond
Dec 3 '18 at 2:38
$begingroup$
We have $c$ be a positive real, I edited the question.
$endgroup$
– elrond
Dec 3 '18 at 2:38
add a comment |
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$begingroup$
Not sure if it matters, but is $c$ a positive real number or a nonzero natural?
$endgroup$
– jgon
Dec 3 '18 at 2:35
$begingroup$
We have $c$ be a positive real, I edited the question.
$endgroup$
– elrond
Dec 3 '18 at 2:38