# Thread: i cant understand this liminf/sup definition

1. ## i cant understand this liminf/sup definition

i got this definition

http://img515.imageshack.us/img515/5666/47016823jz1.gif

what is the meaning of n>=0 under a sup.
sup is not a limit
its only a number
we cant put index under it
what is the meaning of this indexes?

2. Originally Posted by transgalactic
i got this definition

http://img515.imageshack.us/img515/5666/47016823jz1.gif

what is the meaning of n>=0 under a sup.
sup is not a limit
its only a number
we cant put index under it
what is the meaning of this indexes?
Remember that if $x_n$ is bounded then $\limsup x_n = \lim \left( \sup \{ x_k | k\geq n\} \right)$.
The sequence, $\sup \{ x_k | k\geq n\}$ is non-increasing, therefore its limits is its infimum.
Thus, $\limsup x_n = \inf \{ \sup\{ x_k | k\geq n\} | n\geq 0 \}$

3. why you are saying that its not increasing
$

\sup \{ x_k | k\geq n\}
$

its a supremum its only one number its the least upper bound of this sequence
its a constant number it cannot increase or decrease.

and then you are taking the great low bound of this constant number

???

4. Originally Posted by transgalactic
why you are saying that its not increasing
$

$
$\sup \{ x_k | k\geq n\}
" alt="\sup \{ x_k | k\geq n\}
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its a supremum its only one number its the least upper bound of this sequence
its a constant number it cannot increase or decrease.

and then you are taking the great low bound of this constant number

???
If $\{ x_n \}$ is a bounded sequence which is non-increasing i.e. $x_0 \geq x_1 \geq x_2 \geq x_3 ...$
Then $\lim x_n = \inf \{ x_0,x_1, x_2,... \}$.

5. Originally Posted by ThePerfectHacker
If $\{ x_n \}$ is a bounded sequence which is non-increasing i.e. $x_0 \geq x_1 \geq x_2 \geq x_3 ...$
Then $\lim x_n = \inf \{ x_0,x_1, x_2,... \}$.
ok i understood the last inf putting step.

but why you are saying that
$\sup \{ x_k | k\geq n\}$
is not increasing.
you are taking a bounded sequence and you get one number
which is SUP (its least upper bound)
thats it
no more members

??