How to show Sup over bigger set is bigger and Inf over bigger set is lesser?
up vote
0
down vote
favorite
Let $A,B in mathbb{R}^n$ be two subset such that $A subseteq B$. Also, let $f:mathbb{R}^n rightarrow mathbb{R}$ be a real-valued function.
I always use
$$sup_Af(x) leq sup_Bf(x)$$
That perfectly makes sense to me, but is there any proof for the above inequality?
Also for the following
$$inf_Bf(x) leq inf_Af(x)$$
real-analysis functions supremum-and-infimum
asked Nov 25 at 19:08
Saeed
465110
add a comment |
up vote
0
down vote
favorite
Let $A,B in mathbb{R}^n$ be two subset such that $A subseteq B$. Also, let $f:mathbb{R}^n rightarrow mathbb{R}$ be a real-valued function.
I always use
$$sup_Af(x) leq sup_Bf(x)$$
That perfectly makes sense to me, but is there any proof for the above inequality?
Also for the following
$$inf_Bf(x) leq inf_Af(x)$$
real-analysis functions supremum-and-infimum
asked Nov 25 at 19:08
Saeed
465110
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $A,B in mathbb{R}^n$ be two subset such that $A subseteq B$. Also, let $f:mathbb{R}^n rightarrow mathbb{R}$ be a real-valued function.
I always use
$$sup_Af(x) leq sup_Bf(x)$$
That perfectly makes sense to me, but is there any proof for the above inequality?
Also for the following
$$inf_Bf(x) leq inf_Af(x)$$
real-analysis functions supremum-and-infimum
asked Nov 25 at 19:08
Saeed
465110
Let $A,B in mathbb{R}^n$ be two subset such that $A subseteq B$. Also, let $f:mathbb{R}^n rightarrow mathbb{R}$ be a real-valued function.
I always use
$$sup_Af(x) leq sup_Bf(x)$$
That perfectly makes sense to me, but is there any proof for the above inequality?
Also for the following
$$inf_Bf(x) leq inf_Af(x)$$
real-analysis functions supremum-and-infimum
real-analysis functions supremum-and-infimum
asked Nov 25 at 19:08
Saeed
465110
asked Nov 25 at 19:08
Saeed
465110
asked Nov 25 at 19:08
Saeed
465110
asked Nov 25 at 19:08
Saeed
465110
asked Nov 25 at 19:08
Saeed
465110
465110
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
Hint: $sup_{xin B}f(x)$ is an upper bound of the set ${f(x),|,xin A}$.
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
add a comment |
up vote
1
down vote
Let $xin A$. Then in particular $xin B$ and $f(x)geq inf_{xin B}f(x)$. So $inf_{xin B}f(x)$ is a lower bound for the values of $f(x)$ on $xin A$. In particular $inf_{xin A}f(x)geq inf_{xin B}f(x)$.
answered Nov 25 at 19:11
Foobaz John
20.2k41250
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint: $sup_{xin B}f(x)$ is an upper bound of the set ${f(x),|,xin A}$.
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
add a comment |
up vote
1
down vote
Hint: $sup_{xin B}f(x)$ is an upper bound of the set ${f(x),|,xin A}$.
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
add a comment |
up vote
1
down vote
up vote
1
down vote
Hint: $sup_{xin B}f(x)$ is an upper bound of the set ${f(x),|,xin A}$.
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
Hint: $sup_{xin B}f(x)$ is an upper bound of the set ${f(x),|,xin A}$.
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
answered Nov 25 at 19:11
José Carlos Santos
144k20113214
144k20113214
add a comment |
add a comment |
up vote
1
down vote
Let $xin A$. Then in particular $xin B$ and $f(x)geq inf_{xin B}f(x)$. So $inf_{xin B}f(x)$ is a lower bound for the values of $f(x)$ on $xin A$. In particular $inf_{xin A}f(x)geq inf_{xin B}f(x)$.
answered Nov 25 at 19:11
Foobaz John
20.2k41250
add a comment |
up vote
1
down vote
Let $xin A$. Then in particular $xin B$ and $f(x)geq inf_{xin B}f(x)$. So $inf_{xin B}f(x)$ is a lower bound for the values of $f(x)$ on $xin A$. In particular $inf_{xin A}f(x)geq inf_{xin B}f(x)$.
answered Nov 25 at 19:11
Foobaz John
20.2k41250
add a comment |
up vote
1
down vote
up vote
1
down vote
Let $xin A$. Then in particular $xin B$ and $f(x)geq inf_{xin B}f(x)$. So $inf_{xin B}f(x)$ is a lower bound for the values of $f(x)$ on $xin A$. In particular $inf_{xin A}f(x)geq inf_{xin B}f(x)$.
answered Nov 25 at 19:11
Foobaz John
20.2k41250
Let $xin A$. Then in particular $xin B$ and $f(x)geq inf_{xin B}f(x)$. So $inf_{xin B}f(x)$ is a lower bound for the values of $f(x)$ on $xin A$. In particular $inf_{xin A}f(x)geq inf_{xin B}f(x)$.
answered Nov 25 at 19:11
Foobaz John
20.2k41250
answered Nov 25 at 19:11
Foobaz John
20.2k41250
answered Nov 25 at 19:11
Foobaz John
20.2k41250
answered Nov 25 at 19:11
Foobaz John
20.2k41250
20.2k41250
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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%2f3013245%2fhow-to-show-sup-over-bigger-set-is-bigger-and-inf-over-bigger-set-is-lesser%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
