I think that before anyone can tell you why those "are not elements of W", you will have tell us what W is! How is "W" defined?
Folks, Simple questions.
Why are these not elements of W?
(1,2,3,....1000,0,0,0). The finite numbers stop at 1000 and the rest are 0, hence is it not a element of W?
(1,1,1....1, 0,0,0) Not sure of this one?
With (1, 1, 1, ..., 1, 0, 0, 0) you are apparently not told exactly how many "1"s there are but you should be able to see that, whatever that number is, it is a number and so finite.(1,1,1....1, 0,0,0) Not sure of this one?
are these printed lecture notes? is there more text surrounding these statements? context is important. a symbol like:
(1,1,1,....,1,0,0,0) could mean many things.
the presence of a right parenthesis suggests this is not an infinite sequence, as infinite sequences do not terminate.
even if "what was meant" was: an infinite number of 1's and then 3 zeros, that is still not an infinite sequence.
an infinite sequence is just a function where X is some set (the set the terms of the sequence is in).
the trouble with saying (1,1,1,....,1,0,0,0) is an infinite sequence is: which natural numbers are the pre-images of the last 3 terms?
we cannot decide, so the sequence is ill-defined (you can't count up to infinity, and then count 3 more, infinity + 3 is NOT a natural number).
we still have a problem determining WHEN the entries turn from 1's to 0. assuming this is NOT a "sequence of sequences", it would appear that we have an element of W, but it's not a very precise way of indicating it. to specify a sequence, you should be able (in theory, at least) specify every term.
for example, in your last example, the explicit function was:
f(n) = n, n ≤ 1000
f(n) = 0, n > 1000
with (1,1,1,....,0,0,0,....) how can we say what the 1000th term is, and-more importantly for deciding whether or not we have an element of W-how can we say for sure that there is some finite positive integer k, for which f(n) = 0 for all n > k?
it's not "good enough" to say, well, eventually, every term is 0, unles you specifically say: after THIS number, every term is 0. this goes back to the same problem we have before: we can't just put in "infinite ones" and then say we have "zeros after". that's not a well-defined function on the natural numbers.
so again, what is MEANT by (1,1,1,...,0,0,0,.....). i am not going to be sure it's actually an infinite sequence until i know what's in the first "...." (finite? all ones? something i don't know what the heck?).