How do I show that $text{dist}(x,A)=varepsilon + text{dist}(x,B_{varepsilon}(A))$, if $text{dist}(x,A)gt...
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Let $(X,d)$ be a metric space and $Asubset X$ a non-empty subset. Define $B_{varepsilon}(A):={y in X: text{dist}(y,A)leqvarepsilon}$. I want to prove that if $xin X$ with $text{dist}(x,A)gt varepsilon$, then
$$text{dist}(x,A)=varepsilon + text{dist}(x,B_{varepsilon}(A)).$$
I already showed $text{dist}(x,A)leqvarepsilon + text{dist}(x,B_{varepsilon}(A))$, but I struggle showing $text{dist}(x,A)geqvarepsilon + text{dist}(x,B_{varepsilon}(A)).$
calculus general-topology metric-spaces
asked Nov 21 at 11:32
donhansis
11
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up vote
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Let $(X,d)$ be a metric space and $Asubset X$ a non-empty subset. Define $B_{varepsilon}(A):={y in X: text{dist}(y,A)leqvarepsilon}$. I want to prove that if $xin X$ with $text{dist}(x,A)gt varepsilon$, then
$$text{dist}(x,A)=varepsilon + text{dist}(x,B_{varepsilon}(A)).$$
I already showed $text{dist}(x,A)leqvarepsilon + text{dist}(x,B_{varepsilon}(A))$, but I struggle showing $text{dist}(x,A)geqvarepsilon + text{dist}(x,B_{varepsilon}(A)).$
calculus general-topology metric-spaces
asked Nov 21 at 11:32
donhansis
11
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $(X,d)$ be a metric space and $Asubset X$ a non-empty subset. Define $B_{varepsilon}(A):={y in X: text{dist}(y,A)leqvarepsilon}$. I want to prove that if $xin X$ with $text{dist}(x,A)gt varepsilon$, then
$$text{dist}(x,A)=varepsilon + text{dist}(x,B_{varepsilon}(A)).$$
I already showed $text{dist}(x,A)leqvarepsilon + text{dist}(x,B_{varepsilon}(A))$, but I struggle showing $text{dist}(x,A)geqvarepsilon + text{dist}(x,B_{varepsilon}(A)).$
calculus general-topology metric-spaces
asked Nov 21 at 11:32
donhansis
11
Let $(X,d)$ be a metric space and $Asubset X$ a non-empty subset. Define $B_{varepsilon}(A):={y in X: text{dist}(y,A)leqvarepsilon}$. I want to prove that if $xin X$ with $text{dist}(x,A)gt varepsilon$, then
$$text{dist}(x,A)=varepsilon + text{dist}(x,B_{varepsilon}(A)).$$
I already showed $text{dist}(x,A)leqvarepsilon + text{dist}(x,B_{varepsilon}(A))$, but I struggle showing $text{dist}(x,A)geqvarepsilon + text{dist}(x,B_{varepsilon}(A)).$
calculus general-topology metric-spaces
calculus general-topology metric-spaces
asked Nov 21 at 11:32
donhansis
11
asked Nov 21 at 11:32
donhansis
11
asked Nov 21 at 11:32
donhansis
11
asked Nov 21 at 11:32
donhansis
11
asked Nov 21 at 11:32
donhansis
11
11
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add a comment |
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You have difficulty with this because it is false! Let $X=mathbb N$ with the discrete metric. Let $A={1}$. Then $B_{epsilon}(A)=A$ for $0<epsilon <1$. Take $x=2$ to get a counterexample.
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
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1 Answer
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1 Answer
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up vote
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You have difficulty with this because it is false! Let $X=mathbb N$ with the discrete metric. Let $A={1}$. Then $B_{epsilon}(A)=A$ for $0<epsilon <1$. Take $x=2$ to get a counterexample.
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
add a comment |
up vote
2
down vote
You have difficulty with this because it is false! Let $X=mathbb N$ with the discrete metric. Let $A={1}$. Then $B_{epsilon}(A)=A$ for $0<epsilon <1$. Take $x=2$ to get a counterexample.
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
add a comment |
up vote
2
down vote
up vote
2
down vote
You have difficulty with this because it is false! Let $X=mathbb N$ with the discrete metric. Let $A={1}$. Then $B_{epsilon}(A)=A$ for $0<epsilon <1$. Take $x=2$ to get a counterexample.
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
You have difficulty with this because it is false! Let $X=mathbb N$ with the discrete metric. Let $A={1}$. Then $B_{epsilon}(A)=A$ for $0<epsilon <1$. Take $x=2$ to get a counterexample.
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
answered Nov 21 at 11:53
Kavi Rama Murthy
47.8k31854
47.8k31854
add a comment |
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