Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

On Mon, 11 Feb 2008 16:30:40 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:


Confusion all over the place!

Confusion in math can be avoided by stating clear definitions (first)
and (then) formulating proofs _based on that definitions_.

So it's high time to introduce the notion of limes for sequences of sets
explicitly (due to Paul R. Halmos); I hope Jesse F. Hughes will like it.

"If (E_n) is a sequence of sets, the set E* of all those elements which
belongs to E_n for infinitely many values of n is called the /superior
limit/ of the sequence and is denoted by

E* = lim sup E_n.
n

The set E of all those elements which belongs to E_n for all but
*
a finite number of values of n is called the /inferior limit/ of the
sequence and is denoted by

E* = lim sup E_n.
n

If it happens that E* = E ,
*

we shall use the notation

lim E_n
n

for this set." (Halmos, slightly altered)


Let (E_n) be a sequence of sets. (n e N)

The same in the language of set theory:

We write

E* = lim sup E_n
n
with
x e E* <-> {n e N : x e E_n} infinite

We write

E = lim inf E_n
* n
with
x e E <-> {n e N : x !e E_n} finite
*

And if E* = E , we write
*

lim E_n.
n

for this set (and we say that the limit exists).

This definitions are "compatible" (in a certain sense) with the one
Jesse F. Hughes used so far. Since we can rewrite them the following
way:

x e E* <-> An e N Em e M: m >= n & x e E_m,

x e E <-> En e N Am e N: m >= n -> x e E_m.
*

For simplicity we assume that n, m are in N, then we can write:

x e E* <-> AnEm >= n: x e E_m,

x e E <-> EnAm >= n: x e E_m.
*

Thinking about it, I get the following criteria for the limit - if it
exists:

E = lim E_n
n
iff
x e E <-> EnAm >= n: x e E_m for all x.

With other words, x is element of the limit (if it exists) iff it is not
element _only_ of finitely many elements E_n.

This shows that the definitions mentioned above are reasonable. In a
certain sense E_n is "approaching" the limit (if it exists), just like
in the case of the usual limit for numbers (in analysis).

For example:

lim A_n = N
n
with
A_n := {m e N : m < n}.

A_0 = {}
A_1 = {0}
A_2 = {0,1}
A_3 = {0,1,2}
:

N = {0,1,2,...}


F.

--

E-mail: info<at>simple-line<dot>de
.

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