for bounded series $x_n$, prove that $limsup($1/x_n$) = 1/ liminf( $x_n$) $
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stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,
a and b are real numbers (not infinite)
and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive
how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $
i know that $limsup(x_n) = -liminf(- $x_n$) $
and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.
thanks in advance!
calculus sequences-and-series supremum-and-infimum limsup-and-liminf
asked 2 days ago
Boaz Yakubov
204
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up vote
0
down vote
favorite
stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,
a and b are real numbers (not infinite)
and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive
how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $
i know that $limsup(x_n) = -liminf(- $x_n$) $
and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.
thanks in advance!
calculus sequences-and-series supremum-and-infimum limsup-and-liminf
asked 2 days ago
Boaz Yakubov
204
What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago
This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago
Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,
a and b are real numbers (not infinite)
and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive
how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $
i know that $limsup(x_n) = -liminf(- $x_n$) $
and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.
thanks in advance!
calculus sequences-and-series supremum-and-infimum limsup-and-liminf
asked 2 days ago
Boaz Yakubov
204
stuck on a proof about series:
i know that $ 0 < a ≤$ $x_n$ $ ≤ b $,
a and b are real numbers (not infinite)
and from this it follows that $1/x_n$ is also bounded by 1/b and 1/a are positive
how can i prove that $ limsup( 1/ $x_n$ ) = 1 / liminf($x_n$) $
i know that $limsup(x_n) = -liminf(- $x_n$) $
and by bolzano-weirstraaus every bounded set has a converging subset..
but i cant seem to think of a solution to this one.
thanks in advance!
calculus sequences-and-series supremum-and-infimum limsup-and-liminf
calculus sequences-and-series supremum-and-infimum limsup-and-liminf
asked 2 days ago
Boaz Yakubov
204
asked 2 days ago
Boaz Yakubov
204
asked 2 days ago
Boaz Yakubov
204
asked 2 days ago
Boaz Yakubov
204
asked 2 days ago
Boaz Yakubov
204
204
What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago
This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago
Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday
add a comment |
What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago
This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago
Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday
What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago
What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago
This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago
This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago
Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday
Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday
add a comment |
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What has this to do with series? There are no series in the statement of the problem.
– José Carlos Santos
2 days ago
This is true if and only if ${x_n}$ converges.
– Yiorgos S. Smyrlis
2 days ago
Nevermind, the question is a duplicate of this one: math.stackexchange.com/questions/1164104/… Thanks anyway
– Boaz Yakubov
yesterday