1. ## 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??

2. 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 $\displaystyle x_n = \tfrac{n}{n+1}$.

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

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

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