[Haskell-cafe] Re: 0/0 > 1 == False

[Jonathan Cast]

On 12 Jan 2008, at 3:33 AM, Cristian Baboi wrote:

> On Sat, 12 Jan 2008 13:23:41 +0200, Kalman Noel  
> <kalman.noel at bluebottle.com> wrote:
>
>> Achim Schneider wrote:
>>> Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
>>> the anything is defined to one (or, rather, is _one_ anything) to be
>>> able to use the abstraction. It's a bit like the difference between
>>> eight pens and a box of pens. If someone knows how to properly
>>> formalise n = 1, please speak up.
>>
>> Sorry if I still don't follow at all.  Here is how I understand  
>> (i. e.
>> have learnt) lim notation, with n ∈ N, a_n ∈ R.  (Excuse my poor
>> terminology, I have to translate this in my mind from German maths
>> language ;-).  My point of posting this is that I don't see how to
>> accommodate the lim notation as I know it with your term. The  
>> limit of
>> infinity?  What is the limit of infinity, and why should I  
>> multiplicate
>> it with 0?  Why should I get 1?
>>
>>     (1) lim a_n  = a                (where a ∈ R)
>>     (2) lim a_n  = ∞
>>     (3) lim a_n  = − ∞
>>     (4) lim { x → x0 } f(x) = y     (where f is a function into R)
>>
>>     (1) means that the sequence of reals a_n converges towards a.
>>
>>     (2) means that the sequence does not converge, because you can
>>         always find a value that is /larger/ than what you hoped  
>> might
>>         be the limit.
>>    (3) means that the sequence does not converge, because you can
>>         always find a value that is /smaller/ than what you hoped  
>> might
>>         be the limit.
>>
>>     (4) means that for any sequence of reals (x_n ∈ dom f)  
>> converging
>>         towards x0, we have lim f(x_n) = y.  For this equation  
>> again, we
>>         have the three cases above.
>
> Suppose lim a_n = a , lim b_n = b, c_2n = a_n, c_2n+1 = b_n.
>
> What is lim c_n ?

If a = b, lim c_n = a = b.  Otherwise lim c_n is undefined, in the  
sense that there is no number c such that the assertion lim c_n = c  
is true.  The `=' in this assertion isn't an equality; it's just a  
bit of math syntax, used in the special case where it isn't overly  
confusing.

jcc