$sup(lim) le lim(sup)$?
$begingroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.9k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
$endgroup$
add a comment |
$begingroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.9k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
$endgroup$
$begingroup$
You can use the backslash in front of commands like $limsup$limsupand $sin$ so they appear upright.
$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
add a comment |
$begingroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.9k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
$endgroup$
$sup_{x,yin A} lim_{ntoinfty} f_n(x,y) le lim_{ntoinfty} sup_{x,yin A} f_n(x,y)$ ?
Proof) $f_nle sup f_n.\ lim f_nle lim(sup f_n)\ sup(lim f_n)le lim(sup f_n)$
I think that if I want this proof to be correct, I need some conditions.
(such as $lim f_n, lim(sup f_n)$ exists...)
Does $lim_{n to infty} sup_{x in X} f_n(x) = sup_{x in X} lim_{n to infty} f_n(x)$?
I read this question and all the counterexamples in the answers were not enough to dispute sup lim $le$ lim sup (Although they dispute sup lim = lim sup)
In summary, what I like to know is,
1.$sup(lim) le lim(sup)$?(with what condition?)
2.What are the required conditions for it to be correct?
Thank you for reading.
real-analysis
real-analysis
edited Jan 1 at 16:03
A.Γ.
22.9k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
edited Jan 1 at 16:03
A.Γ.
22.9k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
edited Jan 1 at 16:03
A.Γ.
22.9k32656
edited Jan 1 at 16:03
A.Γ.
22.9k32656
edited Jan 1 at 16:03
A.Γ.
22.9k32656
22.9k32656
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
asked Oct 3 '17 at 4:33
ArbitraryArbitrary
1447
1447
$begingroup$
You can use the backslash in front of commands like $limsup$limsupand $sin$ so they appear upright.
$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
add a comment |
$begingroup$
You can use the backslash in front of commands like $limsup$limsupand $sin$ so they appear upright.
$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
$begingroup$
You can use the backslash in front of commands like $limsup$
limsup and $sin$ so they appear upright.$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
You can use the backslash in front of commands like $limsup$
limsup and $sin$ so they appear upright.$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44
add a comment |
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$begingroup$
You can use the backslash in front of commands like $limsup$
limsupand $sin$ so they appear upright.$endgroup$
– Chase Ryan Taylor
Oct 3 '17 at 5:19
$begingroup$
Look for a counterexample. Some situation where the inner sup on $x,y$ varies when $n$ varies.
$endgroup$
– GEdgar
Oct 3 '17 at 13:44