I cant understand this explanation of limsup

[transgalactic]

regarding this definition

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

i was told that
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 \}[/quote]


i can't understand the first part

why he is 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

??

 

[HallsofIvy]

But \left{x_k|k\ge n} is not a single sequence- it is a different sequence for every different n.

For example, if {x_n= (-1)^n/n}= {-1, 1/2, -1/3, 1/4, -1/5, ...} then<br /> sup{x_k|k\ge 1} is the largest of {-1, 1/2, -1/3, 1/4, -1/5, ...} which is 1/2. sup{x_k|k\ge 2} is the largest of {1/2, -1/3, 1/4, -1/5, ...}, again 1/2. sup{x_k|k\ge 3} is the largest of {-1/3, 1/4, -1/5, ...}, which is 1/4. Similarly, sup{x_k|k\ge 4} is also 1/4 but sup{x_k|k\ge 5} is 1/6, etc.