Sometimes the limit of a sequence does not exist. For example, if a_{n} =
(− 1)
^{n}, then
lim_{n→∞}a_{n} does not exist. This is because the terms of the sequence are a distance of 1 apart.
Therefore there can’t exist a single number such that all the terms of the sequence are
ultimately within 1∕4 of that number. The nice thing about limsup and liminf is
that they always exist. First here is a simple lemma and definition. First review
the definition of inf and sup on Page 28 along with the simple properties of these
things.
Definition 1.6.1Denote by
[− ∞, ∞ ]
the real line along with symbols ∞ and −∞. Itis understood that ∞ is larger than every real number and −∞ is smaller thanevery real number. Then if
{An}
is an increasing sequence of points of
[− ∞, ∞]
,
lim_{n→∞}A_{n}equals ∞ if the only upper bound of the set
{An}
is ∞. If
{An }
isbounded above by a real number, then lim_{n→∞}A_{n}is defined in the usual way andequals the least upper bound of
{An}
. If
{An}
is a decreasing sequence of points of
[− ∞,∞ ]
, lim_{n→∞}A_{n}equals −∞ if the only lower bound of the sequence
{An}
is−∞. If
{An }
is bounded below by a real number, then lim_{n→∞}A_{n}is defined in theusual way and equals the greatest lower bound of
{An}
. More simply, if
{An }
isincreasing,
nl→im∞ An ≡ sup {An}
and if
{A }
n
is decreasing then
nl→im∞ An ≡ inf{An }.
Lemma 1.6.2Let
{an}
be a sequence of real numbers and let U_{n}≡
sup
{ak : k ≥ n}
. Then
{Un}
is a decreasing sequence. Also if L_{n}≡ inf
{ak : k ≥ n}
,then
{Ln}
is an increasing sequence. Therefore, lim_{n→∞}L_{n}and lim_{n→∞}U_{n}both exist.
Proof: Let W_{n} be an upper bound for
{ak : k ≥ n}
. Then since these sets are getting
smaller, it follows that for m < n, W_{m} is an upper bound for
{ak : k ≥ n}
. In particular if
W_{m} = U_{m}, then U_{m} is an upper bound for
{ak : k ≥ n}
and so U_{m} is at least as large as U_{n},
the least upper bound for
{ak : k ≥ n}
. The claim that
{Ln}
is decreasing is similar.
■
From the lemma, the following definition makes sense.
Definition 1.6.3Let
{an}
be any sequence of points of
[− ∞, ∞ ]
lim sup an ≡ lim sup{ak : k ≥ n }
n→∞ n→∞
lim nin→f∞ an ≡ nli→m∞ inf{ak : k ≥ n}.
Theorem 1.6.4Suppose
{an}
is a sequenceof real numbers and that
lim sup an
n→ ∞
and
lim inf an
n→ ∞
are both real numbers. Then lim_{n→∞}a_{n}exists if and only if
lim inf an = lim supan
n→ ∞ n→ ∞
and in this case,
lnim→∞ an = lim in→nf∞ an = lim ns→up∞ an.
Proof: First note that
sup {ak : k ≥ n} ≥ inf{ak : k ≥ n}
and so,
lim snu→p∞ an ≡ nli→m∞ sup {ak : k ≥ n}
≥ lim inf{ak : k ≥ n}
n→ ∞
≡ lim ni→nf∞an.
Suppose first that lim_{n→∞}a_{n} exists and is a real number a. Then from the definition of a
limit, there exists N corresponding to ε∕6 in the definition. Hence, if m,n ≥ N,
then
|an − am| ≤ |an − a|+ |a− an| < ε + ε = ε.
6 6 3
From the definition of sup
{ak : k ≥ N}
, there exists n_{1}≥ N such that
sup {ak : k ≥ N} ≤ an1 + ε∕3.
Similarly, there exists n_{2}≥ N such that
inf{ak : k ≥ N } ≥ an2 − ε∕3.
It follows that
sup{ak : k ≥ N }− inf{ak : k ≥ N } ≤ |an − an |+ 2ε< ε.
1 2 3
Since the sequence,
{sup{ak : k ≥ N }}
_{N=1}^{∞} is decreasing and
{inf{ak : k ≥ N}}
_{N=1}^{∞} is
increasing, it follows that
0 ≤ Nlim→∞ sup{ak : k ≥ N }− Nlim→∞ inf{ak : k ≥ N } ≤ ε
Since ε is arbitrary, this shows
lim sup{ak : k ≥ N } = lim inf{ak : k ≥ N} (1.1)
N→ ∞ N →∞
lim (sup {ak : k ≥ N }− inf {ak : k ≥ N}) = 0
N →∞
Since sup
{ak : k ≥ N }
≥ inf
{ak : k ≥ N }
it follows that for every ε > 0, there exists N such
that
sup {ak : k ≥ N }− inf {ak : k ≥ N} < ε,
and for every N,
inf{ak : k ≥ N} ≤ a ≤ sup {ak : k ≥ N }
Thus if n ≥ N,
|a− an| < ε
which implies that lim_{n→∞}a_{n} = a. In case
a = ∞ = Nl→im∞ sup{ak : k ≥ N } = Nlim→∞ inf{ak : k ≥ N}
then if r ∈ ℝ is given, there exists N such that inf
{ak : k ≥ N }
> r which is to say that
lim_{n→∞}a_{n} = ∞. The case where a = −∞ is similar except you use sup
{ak : k ≥ N}
.
■
The significance of limsup and liminf, in addition to what was just discussed, is contained
in the following theorem which follows quickly from the definition.
Theorem 1.6.5Suppose
{an}
is a sequence of points of
[− ∞, ∞ ]
. Let
λ = lim sup an.
n→∞
Then if b > λ, it follows there exists N such that whenever n ≥ N,
an ≤ b.
If c < λ, then a_{n}> c for infinitely many values of n. Let
γ = lim inf an.
n→∞
Then if d < γ, it follows there exists N such that whenever n ≥ N,
an ≥ d.
If e > γ, it follows a_{n}< e for infinitely many values of n.
The proof of this theorem is left as an exercise for you. It follows directly from the
definition and it is the sort of thing you must do yourself. Here is one other simple
proposition.
Proposition 1.6.6Let lim_{n→∞}a_{n} = a > 0. Then
lim snu→p∞ anbn = a lim snu→p∞ bn.
Proof: This follows from the definition. Let λ_{n} = sup
{akbk : k ≥ n}
. For all n large
enough, a_{n}> a − ε where ε is small enough that a − ε > 0. Therefore,
λn ≥ sup {bk : k ≥ n}(a− ε)
for all n large enough. Then
lim sup anbn = lim λn ≡ lim sup anbn
n→∞ n→ ∞ n→ ∞
≥ nl→im∞ (sup {bk : k ≥ n}(a− ε))
= (a − ε) lim sup b
n→ ∞ n
Similar reasoning shows
lim sup anbn ≤ (a+ ε)lim sup bn
n→ ∞ n→ ∞
Now since ε > 0 is arbitrary, the conclusion follows. ■