Liminf and Limsup for Bounded Sequences of Real Numbers - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


LIMINF and LIMSUP
for bounded sequences of real numbers
Denitions
Let (c
n
)
1
n=1
be a bounded sequence of real numbers. Dene
a
n
= inffc
k
:kng ; and
b
n
= supfc
k
:kng :
The sequence (a
n
) is bounded and increasing, so it has a limit; call ita. This
limit is by denition the liminf of the sequence (c
n
),
lim inf
n
c
n
= lim
n!1
inffc
k
:kng :
Similarly, the sequence (b
n
) is bounded and decreasing, so it has a limit; call
it b. This limit is by denition the limsup of the sequence (c
n
),
lim sup
n
c
n
= lim
n!1
supfc
k
:kng :
We have for all n that [a
n
;b
n
] contains [a
n+1
;b
n+1
]. The intersection of these
nested intervals is [a;b].
The limit of a bounded sequence need not exist, but the liminf and limsup
of a bounded sequence always exist as real numbers.
When there's no loss of clarity, we might omit the subscript variable (above,
it is n). There are also shorter notations meaning the same thing: lima
n
means lim supa
n
and lima
n
means lim infa.
Associated facts
We continue with the above notations, and also let d
n
denote a second
bounded sequence of real numbers.
1. [a
n
;b
n
] is the smallest closed interval which containsfc
k
:kng .
2. limc
n
exists if and only if limc
n
= limc
n
. In this case limc
n
= limc
n
=
limc
n
.
3. There is a subsequence of (c
n
) which converges to limc
n
.
4. There is a subsequence of (c
n
) which converges to limc
n
.
5. lim(c
n
+d
n
) (limc
n
) + (limd
n
)
(and strict inequality is possible).
Page 2


LIMINF and LIMSUP
for bounded sequences of real numbers
Denitions
Let (c
n
)
1
n=1
be a bounded sequence of real numbers. Dene
a
n
= inffc
k
:kng ; and
b
n
= supfc
k
:kng :
The sequence (a
n
) is bounded and increasing, so it has a limit; call ita. This
limit is by denition the liminf of the sequence (c
n
),
lim inf
n
c
n
= lim
n!1
inffc
k
:kng :
Similarly, the sequence (b
n
) is bounded and decreasing, so it has a limit; call
it b. This limit is by denition the limsup of the sequence (c
n
),
lim sup
n
c
n
= lim
n!1
supfc
k
:kng :
We have for all n that [a
n
;b
n
] contains [a
n+1
;b
n+1
]. The intersection of these
nested intervals is [a;b].
The limit of a bounded sequence need not exist, but the liminf and limsup
of a bounded sequence always exist as real numbers.
When there's no loss of clarity, we might omit the subscript variable (above,
it is n). There are also shorter notations meaning the same thing: lima
n
means lim supa
n
and lima
n
means lim infa.
Associated facts
We continue with the above notations, and also let d
n
denote a second
bounded sequence of real numbers.
1. [a
n
;b
n
] is the smallest closed interval which containsfc
k
:kng .
2. limc
n
exists if and only if limc
n
= limc
n
. In this case limc
n
= limc
n
=
limc
n
.
3. There is a subsequence of (c
n
) which converges to limc
n
.
4. There is a subsequence of (c
n
) which converges to limc
n
.
5. lim(c
n
+d
n
) (limc
n
) + (limd
n
)
(and strict inequality is possible).
6. lim(c
n
+d
n
) (limc
n
) + (limd
n
)
(and strict inequality is possible).
7. If  is a nonnegative real number,
then lim(c
n
) = limc
n
and lim(c
n
) = limc
n
8. lim(c
n
) = limc
n
and lim(c
n
) = limc
n
9. (Putting the last two facts together:)
If  is a negative real number,
then lim(c
n
) = limc
n
and lim(c
n
) = limc
n
For an example of strict inequality in facts 5 and 6, consider (c
n
) =
0; 1; 0; 1; 0; 1; 0::: (1 at even indices) and (d
n
) = 1; 0; 1; 0; 1; 0; 1; 0::: (1 at odd
indices).
Fact 8 is coming from the following fact for a bounded setS of real numbers,
withS denotingfs :s2Sg:
inf(S) = supS and sup(S) = infS :
To get a feel for this, draw S andS for some example intervals: S =
[3;1], S = [3; 1], S = [1; 3]. Check the infs and sups of S andS.
General sequences of real numbers.
Now drop the assumption that the sequence (c
n
) must be bounded. Use
the same denitions for liminf and limsup, but with b
n
= +1 iffc
k
:kng
is not bounded above, and with a
n
=1 iffc
k
: k ng is not bounded
below. (Note, limc
n
=1 i (c
n
) is not bounded below, and limc
n
=1
i (c
n
) is not bounded above.) Now liminf and limsup are dened for all
sequences of real numbers! Moreover, the \associated facts" still are true
after just a little adjustment.
For fact 1, replace [a
n
;b
n
] with [a
n
;b
n
]\R (in other words, if one of the
endpoints is an innity, remove it from [a
n
;b
n
]).
Generalize the interpretation of \limit exists" (converges) to include the
cases where the limit is1 or1 (where we have already a denition of
what e.g. limc
n
=1 means).
For facts 5 and 6, generalize \+" to include, for every real number :
1+ =1;1+ =1. Also dene1+1 =1 and1+(1) =1.
With these denitions, the facts 5 and 6 are still true. The one thing we don't
do is dene1 +1. Knowing only limc
n
=1 and limd
n
=1 gives us
zero information about lim(c
n
+d
n
) or lim(c
n
+d
n
).