Let s n be a bounded sequence of real numbers. i. Show that for any subsequence s n k of s n , we have lim sup s n k ≤ lim sup s n . Conclude that if a is any accumulation point of s n , then a ≤ lim...

Lets

_n
 be a bounded sequence of real numbers.
i. Show that for any subsequences

_n
k
 ofs

_n
, we have lim sups

_n
k
 ≤ lim sups

_n
. Conclude that ifa is any accumulation point ofs

_n
, thena ≤ lim sups

_n
.
ii. Show that for anyε, there are infinitely many values ofs

_n
 withinε of lim sups

_n
. Conclude that lim sups

_n
 is an accumulation point ofs

_n
.


Hint for ii: One way to proceed is to consider separately the case where there are infinitely many points at leastε greater than the lim sup, and the case where all but finitely many points are at leastε smaller than the lim sup.