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
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