What Is the New Definition of Bolzano-Weierstrass Law Regarding lim inf?

[transgalactic]

if a sequence Y_n is bounded
then there is sub sequence Y_r_n which satisfies
lim inf Y_n<=lim Y_r_n
as n->+infinity

i didnt here of that definition before
the only definition i know about liminf is that it the supremum of all the infimums of the sequence

using the definition i know

this one

lim inf Y_n<=lim Y_r_n
as n->+infinity

doesnt make any sense
??

 

[HallsofIvy]

The usual statement of the Bolzano-Weierstrasse property is that any bounded sequence has a convergent subsequence. In order that you can talk about "lim Y_r_j" that must be true. Since there exist at least one convergent subsequenc, the set of subsequential limits is non-empty and since the sequence itself is bounded the set of subsequential limits is bounded and has both sup and inf. That is, liminf and limsup for the sequence exist as finite numbers (if {a_n} is unbounded below, some texts say the liminf is "negative infinity" in order to always have a liminf). Since the limit of the sequence Y_r_j is one of the subsequential limits, obviously it must be larger than or equal to the liminf, which is a lower bound on the subsequential limits.

 

[transgalactic]

thanks i understood that

so one the one hand liminf is the largest lower bound of all the limits of all the subsequences
and on the other hand its the supremum of the infimums of every subsequence

on a bounded sequence.

 

[HallsofIvy]

"Infimum" is, by definition, the largest of all lower bounds. I'm not sure I like the phrasing "supremum of the infimums of every subsequence" but I think that is the basic idea.