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
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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
61
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donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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up vote
1
down vote
favorite
up vote
1
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
61
New contributor
donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
61
New contributor
donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 21 at 11:32
donhansis
61
New contributor
donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Nov 21 at 11:32
donhansis
61
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donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Nov 21 at 11:32
donhansis
61
asked Nov 21 at 11:32
donhansis
61
61
New contributor
donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
donhansis is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1 Answer
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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
42.4k31751
add a comment |
1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
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
42.4k31751
add a comment |
up vote
3
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
42.4k31751
add a comment |
up vote
3
down vote
up vote
3
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
42.4k31751
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
42.4k31751
answered Nov 21 at 11:53
Kavi Rama Murthy
42.4k31751
answered Nov 21 at 11:53
Kavi Rama Murthy
42.4k31751
answered Nov 21 at 11:53
Kavi Rama Murthy
42.4k31751
42.4k31751
add a comment |
add a comment |
donhansis is a new contributor. Be nice, and check out our Code of Conduct.
donhansis is a new contributor. Be nice, and check out our Code of Conduct.
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