Showing a sequence is bounded and convergent to its infimum.

[ryanj123]

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.

 

[lanedance]

say your sequence is {x_i}

non-increasing means
x_i - x_j <0 for all i>j
which is monotonic

bounded from below means
there exists a lower bound a, such that for all i, x_i > 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{x_i) (=inf{x_i})

 

[ryanj123]

say your sequence is {x_i}

non-increasing means
x_i - x_j <0 for all i>j
which is monotonic

bounded from below means
there exists a lower bound a, such that for all i, x_i > 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{x_i) (=inf{x_i})

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...

 

[lanedance]

so as the sequence is bounded below, you know it has a glb = inf{x_i}

by definition of infinium

say X = {x_i}
a = inf{x_i}

then for any x_i in X , then there exists x_j in X such that
a < x_j < x_i

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

what is your defintion of convergence?