LIM SUP AND LIM INF
DANIEL S. FREED
Let p : Zą0 Ñ R be a sequence of real numbers. (We usually write tpn u Ă R, but we must
keep in mind that the sequence p1 , p2 , p3 , . . . may have repetitions, i.e., the function p need not be
injective. We sometimes also denote the range of p as tpn u.) Then lim supnÑ8 pn and lim inf nÑ8 pn
are defined in the extended real numbers, which are the real numbers with `8 and ´8 adjoined.
So we must make ammendments to some definitions to incorporate these infinities.
For example, if E Ă R is a nonempty subset, then we defined sup E P R only in the case that
E is bounded above. If we work in the extended reals we can define sup E for every nonempty
set by setting sup E “ `8 if E is not bounded above. Similarly, if E is not bounded below we
set inf E “ ´8. If q : Zą0 Ñ R is a sequence such that for every C P Rą0 there exists N P Zą0
such that qn ą C if n ě N , then we write limnÑ8 qn “ `8. Similarly, if for every C P Rą0 there
exists N P Zą0 such that qn ă ´C if n ě N , then we write limnÑ8 qn “ ´8.
Here is a definition of lim sup and lim inf which justifies the nomenclature.
Definition 1. Let p : Zą0 Ñ R be a sequence of real numbers. Then
„
(2)
lim sup pn “ lim
nÑ8

sup pm
nÑ8 měn
and
„
(3)
lim inf pn “ lim
nÑ8

inf pm
nÑ8 měn
In (2) we have the monotonically nonincreasing sequence q defined by
(4)
qn “ sup pm ,
měn
n P Zą0 ,
of extended real numbers which, by monotonicity, necessarily has a limit in the extended real
numbers. Similarly, the sequence r defined by
(5)
rn “ inf pm ,
měn
n P Zą0 ,
is a monotonically nondecreasing sequence of extended real numbers.
Date: May 12, 2016.
1
2
D. S. FREED
Example 6. Consider the following sequences:
(7)
(8)
(9)
(10)
1,
2,
3,
4 , ...
´1, ´2 , ´3 , ´4 , . . .
1, ´1 ,
1
1,
,
2
1 , ´1 , . . .
1
1
,
, ...
3
4
For (7) the sequence (4) of sups is the constant sequence with value `8, so lim sup “ `8. The
sequence (5) of infs equals the sequence (7), and so lim inf “ `8. For (8) the sequence of sups
equals the sequence (8) and the sequence of infs is the constant sequence with value ´8. So
lim sup “ lim inf “ ´8. For (9) the sequence of sups is the constant sequence with value 1 and the
sequence of infs is the constant sequence with value ´1; these then equal the lim sup and lim inf,
respectively. Finally, for (10) we find lim sup “ lim inf “ 0.
Recall that a subsequence of a sequence p : Zą0 Ñ R is a composition
(11)
g
p
Zą0 ÝÝÑ Zą0 ÝÝÑ R
in which g is a strictly increasing function. A subsequential limit is an extended real number which
is the limit of a convergent subsequence.
Lemma 12. Let p : Zą0 Ñ R be a sequence and x P R. Then x is a subsequential limit of p if and
only if for every ą 0 there exist infinitely many n P Zą0 such that pn P px ´ , x ` q.
Proof. If the criterion is satisfied, then we inductively choose for each i P Zą0 a positive integer
gpiq P Zą0 such that |x ´ pgpiq | ă 1{i and gpiq ą gpi ´ 1q. Then the subsequence p ˝ g, usually
written tpgpiq u, converges to x. Conversely, if x is a subsequential limit, say pgpiq Ñ x, then from the
definition of convergence given ą 0 we can find I P Zą0 such that |x ´ pgpiq | ă for all i ě I. Example 13. The sequence (9) has subsequences with value `1 and constant subsequences with
value ´1. An example of the former is gpiq “ 2i ´ 1; an example of the latter is gpiq “ 2i.
Theorem 14. Let p : Zą0 Ñ R be a sequence of real numbers. Set S be the set of subsequential
limits. Then
lim sup pn “ sup S
(15)
nÑ8
lim inf pn “ inf S
nÑ8
Furthermore, tpn u converges if and only if lim sup pn “ lim inf pn , in which case the common value
equals the limit.
Note that S may contain `8 and/or ´8, and also S is never empty. If S contains `8, then
sup S “ `8, and if S contains ´8, then inf S “ ´8.
3
Example 16. For (7), (8), and (10) the set S contains a single element: `8, ´8, and 0, respectively. For (9) we have S “ t`1, ´1u. As a more interesting example, define a sequence tpn u Ă R
by enumerating the rational numbers. Then S “ R, lim sup pn “ `8, and lim inf pn “ ´8.
Remark 17. The intersection S X R is a closed subset of R. One way to see this is to prove directly
from Lemma 12 that
(18)
SXR“
8
č
tpm : m ě nu
n“1
and use the fact that an intersection of closed sets is closed.
Proof of Theorem 14. We prove the statement for lim sup; the lim inf statement follows from
lim infppn q “ ´ lim supp´pn q. If sup S “ `8, then either `8 P S or there exists a sequence
txn u Ă S such that xn Ñ `8. In the former case there is a subsequence pgpiq Ñ `8. It follows
that for each n the set tpm : m ě nu is unbounded above, so the sequence tqn u of supremums
defined in (4) is the constant sequence with value `8. Therefore, lim sup pn “ `8. If there is a
subsequence txn u Ă S converging to `8, then for each C P Rą0 there exists n P Zą0 such that
xn ą C and then, since xn is a subsequential limit, there exists m ě n such that |xn ´pm | ă xn ´C,
which implies pm ą C. It follows again that the sequence (4) of supremums is constant with
value `8.
Thus we are reduced to the case when sup S P R. Note in particular that `8 R S and so tpn u is
bounded above by, say, a positive real number C. If x ą sup S we claim that tm : pm ą xu is
finite. For if not, then tpn u X rx, Cs is infinite, and since rx, Cs is compact there is a convergent
subsequence whose limit is ě x. But then it follows that only finitely many qn in (4) satisfy qn ą x,
and therefore lim sup pn ď x. Since this holds for all x ą sup S we conclude
(19)
lim sup pn ď sup S.
nÑ8
On the other hand, if y ă sup S then there exists x P S such that y ă x ď sup S, and so a
convergent subsequence pgpiq Ñ y. It follows that qn ě y for all n, and thus lim sup pn ě y. Since
this holds for all y ă sup S, we conclude
(20)
lim sup pn ě sup S.
nÑ8
Combine (19) and (20) to deduce lim sup pn “ sup S, as desired.
The argument in the proof characterizes the lim sup.
Corollary 21. Let p : Zą0 Ñ R be a sequence of real numbers. Then if p˚ “ lim sup pn ă `8 we
can characterize p˚ as the unique real number which satisfies: (i) p˚ is a subsequential limit of p,
and (ii) if x ą p˚ then only finitely many pn satisfy pn ě x.
There is a similar characterization of lim inf.