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In classical calculus, a basic concept is a sequence. A sequence is a function from
to . We then define the following:
Limit
The limit of the sequence a_n is x
iff
For every epsilon > 0, there exists an n such that for all m > n, |a_m - x| < epsilon
Subsequence (Ayal's definition)
If
is a sequence, and is a monotone increasing sequence, then is a subsequence . A sequence is said to be a subsequence of if such a monotone increasing sequence exists such that .;Partial limit (also known as subsequential limit)
x is a partial limit of a_n if x is the limit of some subsequence of a_n
Ayal has proven that a sequence can have at most one limit (here: Limits are unique) but some sequences have several partial limits, for example the sequence:
0, 1, 0, 1, 0, 1, ...
has 0 and 1 as partial limits
and the sequence a_n = (-1)^n + 1/n has -1 and 1 as partial limits.
Now a big question.
Question:
Find a sequence whose partial limits are all the elements of
.Ayal's Answer:
1, 2, 3, ..., 1, 2, 3, ..., 1, 2, 3, ..., ...
Now, this is not a sequence, since it is not a function from
to . But Ayal asks, "so what?", it's some sort of sequence, and Ayal suggests that we define a limit of a set, not a sequence. Maybe we don't need sequences at all. If sequences are to be an intuitive concept, then it seems that Ayal's answer should also be a sequence. And if a sequence represents the specific technical mathematical object which is a function from
to , then that's pretty stupid. Let's try without it.Main Question: Can and how would we define the equivalents of limit, subsequence and partial limit in the language of sets instead of sequences?
Ayal realizes that we need multisets and not just sets, since 1,2,1,1,1,1,1, ... and 1,2,1,2,1,2, ... are the same as sets:
ayalgelles: so
ayalgelles: what is the difference between the corresponding set for 1,2,1,2,1,2,1,2,1,2....
and for 1,2,1,1,1,1,1,1,1 ?
me: and for 1,2?
ayalgelles: no, but that is not a sequence
me: why would it not
but if you want ok
ayalgelles: it has to be infinite
me: dont you think thats strange
ayalgelles: what is?
me: that it needs to be infinite?
ayalgelles: well ok.. you said it
me: if you dont want finite sequences
thats ok
so
yues
so you're right
theres some problem here
that sets dont really represent waht we want
because sets dont count multiplicities
at all
and if we want some sort of limit
ayalgelles: yeah so..
me: we need to
so
there's a concept called a multiset
which is like a set
but allow multiplicities
so:
there are 3 basic structures:
(1) set - no order, no multiplicities
(2) multiset - no order, yes multiplicities
(3) sequence - yes order, yes multiplicities
ayalgelles: ohz
so were using multisets?
me: yes we must
ayalgelles: ok
now i will continue to think about it
me: ok cool
http://en.wikipedia.org/wiki/Multiset
Updated Main Question: Can and how would we define the equivalents of limit, subsequence and partial limit in the language of multisets instead of sequences?
Avital: I just realized that multisets by their standard definition only allow finite multiplicities per element, so 1,2,1,2,... is not a multiset by their definition. So in our definition it will. And I don't care right now about the proper mathematical formulation to put it in. It just makes sense, "a collection of elements without order with multiplicities".
---
Here's the definition of limit:
Limit
The limit of the sequence a_n is x
iff
For every epsilon > 0, there exists an n such that for all m > n, |a_m - x| < epsilon
If we try to convert this definition to the language of multisets or sets, we need to re-write "there exists an n such that for all m > n" into something set-theoretical. At first we can re-write the definition of limit as:
Limit
The limit of the sequence a_n is x
iff
For every epsilon > 0, there exists a k such that for all b in {a_m | m > k}, |b - x| < epsilon
This is an equivalent formation to the prior, but slightly more set-theoreical. Now, if we can find what kinds of sets can be {a_m | m > k} for proper {a_n} and proper k. Or more precisely: If I have a set A, what subsets of A can be of the form
{a_m | m > k} for proper choice of sequence {a_n} covering A and k? For example,
If A =
1:20 AM
1:21 AM
1:22 AM
1:23 AM
1:19 AM
, and B = {2, 3, 4, ...}, then we can choose the sequence a_n = n and k = 1.If A = , and B = {2, 4, 5, 6, 7, ...}, then we can choose the sequence a_n = 1, 3, 2, 4, 5, 6, 7, ... and k = 2.If A = , and B = {1, 3, 5, 7, ....}, then we can't do it. So, we are let to the following concept: (starting with
as our sandbox)a subset B of
is called an Oliver set
iff
there exists some sequence {b_n} covering and some k in N such that
B = {b_m | m > k}
We have seen that {2, 3, 4, ...} is an Oliver set, and {2, 4, 5, 6, 7, ...} is an Oliver set and {1, 3, 5, 7, ...} is an Oliver set.
Now, let's find a nice simple amazing characterization of Oliver sets.
It seems that what "keeps a set from being 'oliver'" is that its missing an infinite amount of N elements. This can be wirtten like so: B is an oliver set iff N - B is finite.
4 things to do to complete the definition of the limit of a set:
a proof that indeed a set is oliver iff its complement is finite
complete the definition of a limit of a set (multiset)
prove that it's an extension of the concept of the limit of a sequence, in the following sense:
we need to prove that If {a_n} is a sequence
then lim {a_n} [as a sequence] = lim {a_n} [as a set]
so that indeed we extended the original definition.
Prove that limits are still unique