Why $limsup_{nrightarrowinfty} f_n(x)=inf_k{sup_{ngeq k}f_n}$
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Why
$$limsup_{nrightarrowinfty} f_n(x)=inf_k{sup_{ngeq k}f_n}$$
Somebody thinks it is obviously,.but I can't get it.
Hope your nice answer.
real-analysis
asked Nov 14 at 10:23
Alexander Lau
577
add a comment |
up vote
1
down vote
favorite
Why
$$limsup_{nrightarrowinfty} f_n(x)=inf_k{sup_{ngeq k}f_n}$$
Somebody thinks it is obviously,.but I can't get it.
Hope your nice answer.
real-analysis
asked Nov 14 at 10:23
Alexander Lau
577
There is no need to put "$(x)$" in there. This is a correct formula about a sequence of numbers.
– GEdgar
Nov 14 at 14:33
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Why
$$limsup_{nrightarrowinfty} f_n(x)=inf_k{sup_{ngeq k}f_n}$$
Somebody thinks it is obviously,.but I can't get it.
Hope your nice answer.
real-analysis
asked Nov 14 at 10:23
Alexander Lau
577
Why
$$limsup_{nrightarrowinfty} f_n(x)=inf_k{sup_{ngeq k}f_n}$$
Somebody thinks it is obviously,.but I can't get it.
Hope your nice answer.
real-analysis
real-analysis
asked Nov 14 at 10:23
Alexander Lau
577
asked Nov 14 at 10:23
Alexander Lau
577
asked Nov 14 at 10:23
Alexander Lau
577
asked Nov 14 at 10:23
Alexander Lau
577
asked Nov 14 at 10:23
Alexander Lau
577
577
There is no need to put "$(x)$" in there. This is a correct formula about a sequence of numbers.
– GEdgar
Nov 14 at 14:33
add a comment |
There is no need to put "$(x)$" in there. This is a correct formula about a sequence of numbers.
– GEdgar
Nov 14 at 14:33
There is no need to put "$(x)$" in there. This is a correct formula about a sequence of numbers.
– GEdgar
Nov 14 at 14:33
There is no need to put "$(x)$" in there. This is a correct formula about a sequence of numbers.
– GEdgar
Nov 14 at 14:33
add a comment |
1 Answer
1
active
oldest
votes
up vote
4
down vote
Let $a_k= {sup_ {ngeq k} f_n}$. Verify that $a_{k+1} leq a_k$. For any decreasing sequence $(a_k)$ we have $lim a_n$ and $inf a_k$ are the same. If you change $inf$ on RHS to $lim $ you get LHS.
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Let $a_k= {sup_ {ngeq k} f_n}$. Verify that $a_{k+1} leq a_k$. For any decreasing sequence $(a_k)$ we have $lim a_n$ and $inf a_k$ are the same. If you change $inf$ on RHS to $lim $ you get LHS.
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
add a comment |
up vote
4
down vote
Let $a_k= {sup_ {ngeq k} f_n}$. Verify that $a_{k+1} leq a_k$. For any decreasing sequence $(a_k)$ we have $lim a_n$ and $inf a_k$ are the same. If you change $inf$ on RHS to $lim $ you get LHS.
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
add a comment |
up vote
4
down vote
up vote
4
down vote
Let $a_k= {sup_ {ngeq k} f_n}$. Verify that $a_{k+1} leq a_k$. For any decreasing sequence $(a_k)$ we have $lim a_n$ and $inf a_k$ are the same. If you change $inf$ on RHS to $lim $ you get LHS.
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
Let $a_k= {sup_ {ngeq k} f_n}$. Verify that $a_{k+1} leq a_k$. For any decreasing sequence $(a_k)$ we have $lim a_n$ and $inf a_k$ are the same. If you change $inf$ on RHS to $lim $ you get LHS.
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
answered Nov 14 at 10:33
Kavi Rama Murthy
40.4k31751
40.4k31751
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
add a comment |
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
Thank U ! I can get it !
– Alexander Lau
Nov 14 at 10:43
add a comment |
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There is no need to put "$(x)$" in there. This is a correct formula about a sequence of numbers.
– GEdgar
Nov 14 at 14:33