[Haskell-cafe] Re: 0/0 > 1 == False
Jonathan Cast
jonathanccast at fastmail.fm
Sun Jan 13 19:21:59 EST 2008
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
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