What does All but finitely many n mean?

[kntsy]

what does "All but finitely many n" mean?

Let $\left(s_n\right)$ be a convergent sequence.
If $s_n\geq a$ for all but finitely many n, then $\lim s_n\geq a$.


all n?(all the natural number? but natural number is infinite so why use "but finite"?)
Infinite?
Or finite?



What does it mean actually?

 

[Tac-Tics]

Suppose you have an infinite number of light switches. An infinite number of them are in the off position. What do you know about the number in the on position?

Answer: nothing.

It might be that all the lights are off.

It might be that every other light is on. In this case, there are an infinite number of switches on and and infinite number off.

If might be that only the first five switches are on. In this case, there are only a finite number of switches on.

(This is similar to the concept of a Set of Measure Zero in Measure Theory).

 

[HallsofIvy]

"All but finitely many" says exactly what it means. If a statement is true for "all but finitely many" things (integers, triangles, whatever) then the set of all such things for which it is NOT true is finite.

The statement "n is larger than 10000000000000000" is true for "all but finitely many" positive integers precisely because it if not true for exactly 10000000000000000 positive integers, a finite number. The statement "n is even" is NOT true for "all but finitely many" positive integers because it is not true for all odd positive integers, an infinite set.

 

[statdad]

One more: in relation to a sequence, the statement $s_n \ge a$ for all but finitely many n means that there is some postive integer N such that $s_k \ge a$ for all integers $k \ge N$ (so, eventually, infinitely many terms of the sequence are at least as big as the number $a$.

 

[kntsy]

The statement "n is even" is NOT true for "all but finitely many" positive integers because it is not true for all odd positive integers, an infinite set.

Thanks hallsofivy and others.
but I do not quite understand your statement.
It IS true for "infinite" positive integers(2,4,6,8,10,...100,...1000,...).
Admittedly, it is NOT true for "infinite" positive integers(1,3,......1001...)but why this reason is enough refute the statement?

 

[statdad]

The clause "for all but finitely many" means that the statement in question fails to be true for only finitely many objects. Since there are infinitely many integers that are not even, "n is even for all but finitely many positive integers" is not correct.

 

[julypraise]

English must be your second language I guess. Well, I don't really know, but it's the case for me. And for this very reason, it took some time to get this 'all but finitely many' clearly.

You know.. you should notice that 'all but finitely many' here in this context means not 'all and finitely many' but 'all except finitely many'. And if you interpret that this way, all the explanations should make sense now.

 

[alex_j]

Could it be that it means that the statement holds true for any infinite series but not for a finite series? Since in an infinite series you always have the possibility of 'S' being greater than or equal to 'a' whereas in a finite series 'S' may be less than 'a'?

 

[alex_j]

For example:

Let (S_n) $\geq$a

If (S_n) $\geq$a AND 'n' is infinite we can say that

Lim_n->$\infty$S_n $\geq$ a


For example:
Let S_n = $\sum$1/x_n AND a=2

For an infinite series there is always a possibility for S_n to be $\geq$a but this does not necesarily hold true for a finite series (i.e. n=5,10,1000...etc.)