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From: deltasigma on 16 May 2010 07:57
suppose that we have a sequence of rational approximations a_n of an
irrational number r with a_n<a_(n+1) such that:

a_n<a_(n+1)<a_(n+2)<...<r

What is the lim(n->oo){a_n} a rational or an irrational number?
From: Patrick Coilland on 16 May 2010 09:44
deltasigma a �crit :
> suppose that we have a sequence of rational approximations a_n of an
> irrational number r with a_n<a_(n+1) such that:
>
> a_n<a_(n+1)<a_(n+2)<...<r
>
> What is the lim(n->oo){a_n} a rational or an irrational number?

There are not enough info to answer :
You can have an increasing sequence of "rational approximations" of r
whose limit is any number (rational or irrational) <=r


There is no well known definition for "rational approximations" of r :)


From: HallsofIvy on 16 May 2010 05:53
> suppose that we have a sequence of rational
> approximations a_n of an
> irrational number r with a_n<a_(n+1) such that:
>
> a_n<a_(n+1)<a_(n+2)<...<r
>
> What is the lim(n->oo){a_n} a rational or an
> irrational number?

Just saying "a sequence of rational approximations" is not good enough. Do you mean that the approximations become "arbitrarily close" as n increases? In that case, of course, the limit is just r itself. On the other hand, I could have a sequence that converges to the rational number 1.4142135623730950488016887242097 by defining a_1= 1 and a_{n+1}= (1/2)(1.4142135623730950488016887242097+ a_n). Would you uconsider that " sequence of rational approximations" to sqrt(2)? They come within 30 decimal places of sqrt(2) but no closer.
From: José Carlos Santos on 16 May 2010 10:22
On 16-05-2010 12:57, deltasigma wrote:

> suppose that we have a sequence of rational approximations a_n of an
> irrational number r with a_n<a_(n+1) such that:
>
> a_n<a_(n+1)<a_(n+2)<...<r
>
> What is the lim(n->oo){a_n} a rational or an irrational number?

It depends. If all that you are saying is that the sequence (a_n)_n is
increasing and that each of its terms is smaller than _r_, then its
limit (which must exist), can be any number smaller than or equal
to _r_.

On the other hand, if the word "approximations" means here that
lim_n|a_n - r| = 0, then, of course, the limit of the sequence is _r_,
but then you do not have to assume that the sequence is increasing or
that each of its terms is smaller than or equal to _r_.

Best regards,

Jose Carlos Santos
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