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Showing a sequence is bounded and convergent to its infimum.

  • Thread starter ryanj123
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  • #1
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Homework Statement



Show that any non-increasing bounded from below sequence is convergent to its
infimum.


Homework Equations



Not quite sure... is this a monotonic sequence?

The Attempt at a Solution



At this point I'm not even sure about which route to go. I am in need of serious help.

Thanks.
 

Answers and Replies

  • #2
lanedance
Homework Helper
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say your sequence is {xi}

non-increasing means
xi - xj <0 for all i>j
which is monotonic

bounded from below means
there exists a lower bound a, such that for all i, xi > a

do you have a theorem about a greatest lower bound existing for bounded below sequence? Otherwise you may have to show this exists, then use the monotnic behaviour of the function to show it converges to the glb{xi) (=inf{xi})
 
Last edited:
  • #3
24
0
say your sequence is {xi}

non-increasing means
xi - xj <0 for all i>j
which is monotonic

bounded from below means
there exists a lower bound a, such that for all i, xi > a

do you have a theorem about a greatest lower bound existing for bounded below sequence? Otherwise you may have to show this exists, then use the monotnic behaviour of the function to show it converges to the glb{xi) (=inf{xi})
Are you speaking of the infimum thm.? I know of that. But it's showing the convergence to it that is the problem for me...
 
  • #4
lanedance
Homework Helper
3,304
2
so as the sequence is bounded below, you know it has a glb = inf{xi}

by definition of infinium

say X = {xi}
a = inf{xi}

then for any xi in X , then there exists xj in X such that
a < xj < xi

why? & how does it help...? ;)

what is your defintion of convergence?
 

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