# upper/lower sequence question..

• Jan 8th 2009, 06:11 AM
transgalactic
upper/lower sequence question..
if my sequence is

1/2,2/3,3/4,4/5 ..

why the upper sequence is 1

and lower sequence is 1 too ??

1 is not ever a member of a sequence.

i got the idea that upper sequence is a sub sequence constructed from the highest members
the closest to the upper bound

1 is not even in the sequence??
• Jan 8th 2009, 10:06 AM
ThePerfectHacker
Quote:

Originally Posted by transgalactic
if my sequence is

1/2,2/3,3/4,4/5 ..

why the upper sequence is 1

and lower sequence is 1 too ??

I am assuming this question is about limsup's and liminf's.
Let $x_n = \tfrac{n}{n+1}$.

If $a_n$ is the superior (upper) sequence then $a_n = \sup \{ x_k | k\geq n \} = \sup \left\{ \tfrac{k}{k+1} \bigg| k\geq n \right\} = 1$

If $b_n$ is the inferior (lower) sequence then $b_n = \inf \{ x_k|k\geq n\} = \inf \left\{ \tfrac{k}{k+1} \bigg| k\geq n \right\} = \frac{n}{n+1}$

Therefore, $\limsup x_n = \lim a_n = 1$ and $\liminf x_n = \lim a_n = 1$.