Re: On convergence of sequences of sets. Was Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: G. Frege <nomail@invalid>
- Date: Fri, 15 Feb 2008 18:52:43 +0100
On Wed, 13 Feb 2008 02:00:52 +0000, Gonçalo Rodrigues <nospam@xxxxxxxxxxxx>
wrote:
[Theorem:] Using the above formulas we can readily prove that if the
sequence (E_n) is decreasing than it is convergent and the limit is
/\_n E_n. Similarly, if (E_n) is increasing than it converges to \/_n
E_n.
Halmos:
"If the sequence is such that
E_n ( E_n+1, for n = 1, 2, ...,
it is called /increasing/; if
E_n ) E_n+1, for n = 1, 2, ...,
it is called /decreasing/. Both increasing and decreasing sequences
will be referred to as /monotone/. It is easy to verify that if (E_n)
is a monotone sequence, then lim E_n exists and is equal to
n
_
U E_n or | | E_n
n n
according as the sequence is increasing or decreasing."
"It is easy to verify"? Ok, let's try. (Just for an increasing sequence of
sets.)
Claim:
If E_n c E_n+1 for n = 1, 2, ..., then E^* = U En and E_* = U En.
n n
Hence lim E_n = U E_n.
n n
Proof:
x e E^* -> x e E_n for infinitely many values of n
-> En e N: x e E_n
-> x e U E_n.
n
x e U E_n -> En e N: x e E_n
n
-> Let n0 e N: x e E_n0
-> x e E_n0+1, x e E_n0+2, ... (since E_n c E_n+1 for all n e N)
-> x e E_n for infinitely many values of n
-> x e E^*.
Hence E^* = U E_n.
n
x e E_* -> x !e E_n (only) for finitely many values of n
-> En e N: x e E_n
-> x e U E_n.
n
x e U E_n -> En e N: x e E_n
n
-> Let n0 e N: x e E_n0
-> x e E_n0+1, x e E_n0+2, ... (since E_n c E_n+1 for all n e N)
-> x !e E_n (only) for finitely many values of n
-> x e E_*.
Hence E_* = U E_n. []
n
F.
--
E-mail: info<at>simple-line<dot>de
.
- References:
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: G . Frege
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Han . deBruijn
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Jesse F. Hughes
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Han de Bruijn
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Jesse F. Hughes
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Gonçalo Rodrigues
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Han de Bruijn
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- From: Jesse F. Hughes
- Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
- From: Gonçalo Rodrigues
- On convergence of sequences of sets. Was Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
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