# 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 $\displaystyle x_n = \inf\{ y_k | k\geq n\}$ and $\displaystyle z_n = \sup\{ y_k | k\geq n\}$.
We know that $\displaystyle \lim x_n \leq \lim z_n$.
Let $\displaystyle \{y_{r_n}\}$ be a monotome subsequence.
Then $\displaystyle \{ y_{r_n} | n\geq k\} \subseteq \{ y_n | n\geq k\}$ for all $\displaystyle k\geq 1$.
Therefore, $\displaystyle \sup \{ y_{r_n} | n\geq k\} \geq \sup \{ y_n | n\geq k \}$ and so $\displaystyle z_{r_n} \geq z_n$.
This means, $\displaystyle \lim z_n \leq \lim z_{r_n}$.
Thus, $\displaystyle \lim x_n \leq \lim z_n \leq \lim z_{r_n}$.
By definition this means, $\displaystyle \liminf y_n \leq \limsup y_{r_n}$.
However, $\displaystyle \{y_{r_n}\}$ is monotone and bounded so $\displaystyle \lim y_{r_n}$ exists.
Thus, $\displaystyle \lim y_{r_n} = \limsup y_{r_n}$.
Putting all of this together we get, $\displaystyle \liminf y_n \leq \lim y_{r_n}$.