# Thread: a new definition to bolzano weirshtrass law regarding liminf that i cant understand..

1. ## a new definition to bolzano weirshtrass law regarding liminf that i cant understand..

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
Code:
 lim inf Y_n<=lim Y_r_n as n->+infinity
doesnt make any sense ??

2. Originally Posted by 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
Code:
 lim inf Y_n<=lim Y_r_n as n->+infinity
doesnt make any sense ??
Let $x_n = \inf\{ y_k | k\geq n\}$ and $z_n = \sup\{ y_k | k\geq n\}$.
We know that $\lim x_n \leq \lim z_n$.
Let $\{y_{r_n}\}$ be a monotome subsequence.
Then $\{ y_{r_n} | n\geq k\} \subseteq \{ y_n | n\geq k\}$ for all $k\geq 1$.
Therefore, $\sup \{ y_{r_n} | n\geq k\} \geq \sup \{ y_n | n\geq k \}$ and so $z_{r_n} \geq z_n$.
This means, $\lim z_n \leq \lim z_{r_n}$.
Thus, $\lim x_n \leq \lim z_n \leq \lim z_{r_n}$.
By definition this means, $\liminf y_n \leq \limsup y_{r_n}$.
However, $\{y_{r_n}\}$ is monotone and bounded so $\lim y_{r_n}$ exists.
Thus, $\lim y_{r_n} = \limsup y_{r_n}$.
Putting all of this together we get, $\liminf y_n \leq \lim y_{r_n}$.