CM221A
ANALYSIS I
NOTES ON WEEK 3
UPPER AND LOWER LIMITS
One says that +
∞
is an accumulation point of a sequence
{
a
n
}
if there is a subse
quence of
{
a
n
}
which converges to +
∞
. Similarly,
∞
is an accumulation point
of
{
a
n
}
if there is a subsequence which converges to
∞
. This convention makes
sense, even though
±∞
are not proper numbers.
Deﬁnition.
The largest accumulation point of a sequence
{
a
n
}
is called the upper
limit of
{
a
n
}
and is denoted lim sup
a
n
. The smallest accumulation point of
{
a
n
}
is called the lower limit of
{
a
n
}
and is denoted lim inf
a
n
.
The Bolzano–Weierstrass Theorem implies that, for any bounded sequence, both
upper and lower limits are ﬁnite numbers. If the sequence is unbounded the upper
and lower limits may be
±∞
.
Example.
If
a
n
=
n
then lim sup
a
n
= lim inf
a
n
= +
∞
.
Note that the upper and lower limits always exist, even if the sequence does not
converge.
Exercise.
Prove that a sequence
{
a
n
}
has a limit (ﬁnite or inﬁnite) if and only if
lim sup
a
n
= lim inf
a
n
.
Lemma.
If
r >
0 then lim sup(
r a
n
) =
r
lim sup
a
n
and lim inf(
r a
n
) =
r
lim inf
a
n
.
If
r <
0 then lim sup(
r a
n
) =
r
lim inf
a
n
and lim inf(
r a
n
) =
r
lim sup
a
n
.
Proof.
Let
r
6
= 0. From the algebraic rules for limits it follows that a subsequence
{
a
n
k
}
converges to a limit
a
if and only if
{
r a
n
k
}
converges to the limit
r a
. There
fore
c
is an accumulation point of
{
a
n
}
if and only if
r c
is an accumulation point
of the sequence
{
r a
n
}
. In other words, the accumulation points of the sequence
{
r a
n
}
are obtained from the accumulation points of
{
a
n
}
by multiplying them
by the number
r
. If
r >
0 then the multiplication maps the largest and smallest
accumulation points of
{
a
n
}
into the largest and smallest accumulation points of
{
r a
n
}
. If
r <
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 Spring '09
 Limits, lim, Mathematical analysis, Limit of a sequence, lim sup

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