[Haskell-cafe] Re: Re: 0/0 > 1 == False
Ben Franksen
ben.franksen at online.de
Fri Jan 18 21:31:23 EST 2008
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. 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.
(2) usually rather mean that for each positive limit A there is a number N
such that a_N > A for /all/ n > N.
An analogous definition hold for (3).
Your definition of (2) is usually termed as '(a_n) contains a subsequence
that tends toward +infinity'.
Cheers
Ben+
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