How do I show that $text{dist}(x,A)=varepsilon + text{dist}(x,B_{varepsilon}(A))$, if $text{dist}(x,A)gt...
Multi tool use
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.
add a comment |
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.
add a comment |
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.
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.
Check out our Code of Conduct.
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.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
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 |
1 Answer
1
active
oldest
votes
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.
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007604%2fhow-do-i-show-that-textdistx-a-varepsilon-textdistx-b-varepsilon%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
A4g4WjE0f fGvz BJ0FvQRGJGFC UsZ6ly
