On Sat, 12 Jan 2008 13:23:41 +0200, Kalman Noel
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.
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