Page 337 - DMTH401_REAL ANALYSIS

Unit 28: Sequences of Functions and Littlewood’s Third Principle

Notes
Figure 28.1: A Sequence Bouncing Up and Down

Given an arbitrary sequence {a } of extended real numbers, put
n
s = supa = sup{a , a , a ,...},
1 k 1 2 3
³
k 1
s = supa = sup{a , a , a ,...},
2 k 2 3 4
k 2
³
s = supa = sup{a , a , a ,...},
3 k 3 4 5
k 3
³
and in general,
s = supa = sup{a , a , a ,...}.
n k n n+1 n+2
k n
³
Note that
s ³ s ³ s ³ ··· ³ s ³ s ³···
1 2 3 n n+1
is an non-increasing sequence since each successive s is obtained by taking the supremum of a
n
smaller set of elements. Since {s } is an non-increasing sequence of extended real numbers, the
n
limit lim s exists in  ; in fact,
n
lim s = inf s = inf{s , s , s ,...},
n n n 1 2 3
as can be easily be checked. We define the lim sup of the sequence {a } as
n
lim sup a := inf s = lim s = lim (sup{a , a , a ,...})
n n n n n®¥ n n+1 n+2
Note that the term “lim sup” of {a } fits well because lim sup a is exactly the limit of a sequence
n n
of supremums.

Example: For the sequence a shown in Figure 28.1, we have
n
s = a , s = a , s = a , s = a , s = a ,...,
1 1 2 3 3 3 4 5 5 5
so lim sup a is exactly the limit of the odd-indexed a ’s.
n n
We now define the “lower” or “infimum” limit of an arbitrary sequence {a }. Put
n
 = sup a = inf{a , a , a ,...},
1 k 1 2 3
k 1
³
 = sup a = inf{a , a , a ,...},
2 k 2 3 4
³
k 2
 = sup a = inf{a , a , a ,...},
3 k 3 4 5
³
k 3

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