12/2/2015
set theory - Why do the natural numbers get exhausted in some limits but not in all? - MathOverflow
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Why do the natural numbers get exhausted in some limits but not in all? [closed]
Let (sn ) be the sequence of sets sn
1
tn = {n }
or
n
un = {n }
or
= {n}
vn = {
n
n+1
of natural numbers 1, 2, 3, ... Then the limit is the empty set {}. The sequences of sets
}
have also empty limit sets. This shows that no natural number will remain in all sets of the
sequence. All natural numbers will be exhausted.
But this simple argument appears to fail in cases like
wn = {1n }
(where
n
is an index) or
xn = {
n
n
}
or
0
y n = {n } ?
Of course all limits
are {1}. But why don't the natural numbers get exhausted in these cases?
set-theory
asked Jun 27 at 12:37
Bacarra
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closed as off-topic by Joel David Hamkins, Gerry Myerson, Simon Thomas, Stefan Waldmann,
Stopple Jun 27 at 12:55
This question appears to be off-topic for this site. While what’s on- and off-topic is not always intuitive, you can learn more
about it by reading the help center. The users who voted to close gave this specific reason:
"This question does not appear to be about research level mathematics within the scope defined in the help
center." – Joel David Hamkins, Gerry Myerson, Simon Thomas, Stefan Waldmann, Stopple
If this question can be reworded to fit the rules in the help center, please edit your question or leave a comment.
deleted by Todd Trimble ♦ Jun 27 at 14:16
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I don't understand your question, but I'm pretty sure it's not about mathematical research, which is what the MO
website is for. – Gerry Myerson Jun 27 at 12:47
The question is simply why the sequence { n1 } has an empty limit, i.e., loses all natural numbers, but the
0
sequence { n0 } does not lose all natural numbers, i.e. the limit is not {
Bacarra Jun 27 at 13:00
},
whatever that might mean. –
Is this any different from asking why the sequence 1, 2, 3, … doesn't have a limit, whereas the sequence
1, 1, 1, … does? – Gerry Myerson Jun 27 at 13:02
Yes, it is different because the sequence { n1 } has a limit, namely the empty set. That means that all natural
numbers are exhausted in the limit. Why not in the other case? – Bacarra Jun 27 at 13:14
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