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?
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 $\displaystyle x_n$ is bounded then $\displaystyle \limsup x_n = \lim \left( \sup \{ x_k | k\geq n\} \right)$.
The sequence, $\displaystyle \sup \{ x_k | k\geq n\}$ is non-increasing, therefore its limits is its infimum.
Thus, $\displaystyle \limsup x_n = \inf \{ \sup\{ x_k | k\geq n\} | n\geq 0 \} $
why you are saying that its not increasing
$\displaystyle
\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
???