Re: abundance of irrationals!)

In article <fb701d3c.0504221144.153876e7@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> David Kastrup <dak@xxxxxxx> wrote in message
>
> The relevance of
> E z e {2,4,6,...,2n} : z > Card({2,4,6,...,2n})
> for the set Z = {2,4,6,...} of all positive finite even numbers 2n
> has been denied by you by asserting: Z is not a finite sequence!
>
> This not an argument. The correct argument runs as follows:
> We do not know what the set of even numbers is. All we know is that it
> consists of positive finite even numbers only.
> We know that induction is valid for all finite natural numbers.
> Further we know that each positive finite even number is a member of
> the set N of natural numbers n. Further we know that each n defines a
> finite sequence {1,2,3,...,n}.
> Now we can conclude
> E z e {2,4,6,...,2n} : z > Card({2,4,6,...,2n})
> and from that
> E z larger than the cardinality of the set of all positive finite even
> numbers Z.

That is not how limits work.

Given f(n) > g(n) for all n in N, the best one can insist upon, given
that both limits exist, is that LIM f(n) >= LIM g(n)

Note that (n-1)/n < (n+1)/n for all n, but the limits are equal.
.

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