Statements true for all n Vs. statements true as n->infty

Question

Let P be a statement.

What are the necessary and sufficient conditions for the following statement to be true?

(P is true $\forall n \in \Bbb N$)$\implies$(P is true as n$\to \infty$)

As background for this question, I understand $0 \in \bigcap_1^n(-1/n,1/n)$ $\forall n \in \Bbb N$. However, I am not entirely clear as to why $0 \in \bigcap_1^\infty (-1/n,1/n)$.

After all, $-1/n,1/n \to 0$ as $n \to \infty$, but $(-1/n,1/n)$ does not contain both -1/n and 1/n, so why should $0 \in \bigcap_1^\infty (-1/n,1/n)$?

Also, if $c_n \to 1$ as $n \to \infty$ with $c_n<1$ $\forall n \in \Bbb N$, then is it true that $\bigcup_1^\infty [-c_n,c_n]$ = $(-1,1)$?

Answer 1

The implication from "$\forall n\in\Bbb N$" to "For sufficiently large $n\in\Bbb N$" is always true. Because if it's always true... then it's always true.

The reverse implication, of course, fails to hold.

As for the reason $0\in\bigcap_{n=1}^\infty\left(-\frac1n,\frac1n\right)$, it's easier to think about when any element is in an intersection of any family of sets. When it's a member of all the sets in the family. $0$ has this property, $0\in\left(-\frac1n,\frac1n\right)$ for any $n\in\Bbb N$. Therefore it is in the intersection.

Don't think about unions or intersections are limits. This is the union of a family of sets. Is the family finite? Infinite? Uncountable? Nobody cares, because we have an excellent definition:

Let $I$ be any non-empty index set, and let $A_i$ be a set, for every $i\in I$. We define the following: $$\bigcup_{i\in I}A_i = \{x\mid\exists i\in I.x\in A_i\}\\\bigcap_{i\in I}A_i = \{x\mid\forall i\in I.x\in A_i\}$$

Now $\bigcap_{n=1}^\infty\left(-\frac1n,\frac1n\right)$ is simply the intersection where $I=\Bbb N$, and $A_i=\left(-\frac1n,\frac1n\right)$. It is not the limit of some sequence, and thinking about it as such can be confusing.