# LimSup and LimInf

Discussion in 'Undergraduate Math' started by khosaa, Feb 21, 2011.

1. ### khosaaGuest

Hi,

It seems that there are two common definitions for the terms "Limit
Superior" and "Limit Inferior": one pertains to sequences and the
other to sets.

I am curious if the two defns are in any way related. I can't seem to
find such a connection, but perhaps there is some clever way to cast
the LimSup for a sequence in terms of sets that I'm somehow missing.

Thanks,
Fran

khosaa, Feb 21, 2011

2. ### Arturo MagidinGuest

Given a sequence of real numbers {a_n}, let A_n = (-oo,a_n).

The Limsup A_n = (-oo,limsup a_n), where Limsup is the limit superior
of sets, and limsup is the limit superior of sequences. Likewise,
Liminf A_n = (-oo,liminf a_n).

Arturo Magidin, Feb 21, 2011

3. ### Butch MalahideGuest

I'm not sure that's exactly correct. Suppose a_n = 1/n, A_n = (-oo, 1/
n); then limsup a_n = lim a_n = 0, while Limsup A_n = Lim A_n = (-oo,
0], not (-oo, 0).

As an alternative answer to the OP, if f_n(x) is the characteristic
function of A_n, I believe that limsup f_n(x) is the characteristic
function of Limsup A_n, and liminf f_n(x) is the characteristic
function of Liminf A_n.

Butch Malahide, Feb 21, 2011
4. ### Arturo MagidinGuest

Right; thanks for the correction. Endpoints may or may not be
included; Limsup A_n is of the form (-oo,k) or (-oo,k], and k equals
limsup a_n; and liminf A_n is of the form (-oo,m) or (-oo,m], and m
equals liminf a_n.

Arturo Magidin, Feb 21, 2011
5. ### David C. UllrichGuest

I can imagine at least four different things you might mean by that.
If you say exactly what two definitions you're referring to you're
more likely to get an answer to the question you meant to ask.

David C. Ullrich, Feb 21, 2011
6. ### khosaaGuest

Hi David,

I'll simply state the defns for LimSup that I had in mind (since the
corresponding defns for LimInf should be obvious)

The defn I had in mind for the sequence (a_n) was as follows:
then LimSup (a_n) = Inf (over all natural numbers m) { Sup{a_m,a_m
+1,...}}

The defn I had in mind for the family of sets {A_n} was
LimSup A_n = Intersection for m=1 to +Infinity [ Union for n >= m of
A_n]

Butch and Arturo gave a nice explanation of how the two are related:
For any natural number n, define b_n = Sup{a_n,a_n+1,...}
then we have a_m <= b_n for every m>= n implies
(-00,a_m) subset (-00, b_n) for every m>= n implies
Union from m=n to +infinity (-00,a_m) subset (-00,b_n)
but inf of all b_n can be obtained simply by taking the intersection
of all the (-00,b_n)

I know of only these two defns (which I found in the Harper Collins
dictionary of math). Curious what the others are.

Fran

khosaa, Feb 23, 2011
7. ### khosaaGuest

Hi Arturo,

Thank you very much for the nice explanation. Definitely would not
have seen it on my own. ;>)

Fran

khosaa, Feb 23, 2011
8. ### David C. UllrichGuest

Fine, then - that's more or less what Butch and Arturo thought
what you meant.

(In case you're wondering what I was wondering about: there
are various ways of defining the llim sup of a _sequence_
that refer to various _sets_, for example lim sup a_n is
the largest element of S, where S is the set of all accumulation
points. Or, lim sup a_n = L if and only if the set

David C. Ullrich, Feb 23, 2011