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

[Jonathan Cast]

On 12 Jan 2008, at 3:23 AM, Kalman Noel 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.

That's ok, neither does he.

>   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 ;-).

OK so far.  I actually had a professor in uni who would randomly  
switch to German during lectures; I'll do my best to follow your  
notation :)

>   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,

If you extend your concept of `sequence' to include sequences of  
extended real numbers, the limit of the sequence that is identically  
inf is inf.  Otherwise, the notation isn't really meaningful.

> 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,

To a value in R.  Again, inf is a perfectly well defined extended  
real number, and behaves like any other element of R u {-inf, inf}.   
(Although that structure isn't quite a field --- 0 * inf isn't  
defined, nor is inf - inf).

> 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.

jcc