Prove what elements are in the set So I got the two sets: 
$$ I = \left(-1,1+ \frac1n\right)    $$ 
$$ J = \left[0,1-\frac1n\right]$$
whereas $(a,b)= \{{x \in \mathbb R}: a<x<b\}$ 
and $ [a,b]= \{x \in \mathbb R: a  ≤ x ≤b\}$ 
And I have to find the elements of $$\bigcap\limits_{n\in\mathbb N}I $$ and $$
\bigcup\limits_{n\in\mathbb N}J $$
Now I would say that the intersection of I  is $\{{x \in \mathbb R}:-1<x≤1\}$
and that the union of J is $\{{x \in \mathbb R}:0≤x≤1\}$
Am I correct, and if so - how would you formally solve this? The way I did it was just thinking it through in my head but if I had to come up with a prove I might get stuck.
 A: HINT: to check if some point $x$ belongs to $\bigcap I_n$ or $\bigcup J_n$ observe that
$$x\in\bigcap I_n\iff \forall n\in\Bbb N:x\in I_n$$
and
$$x\in\bigcup J_n\iff \exists n\in\Bbb N:x\in J_n$$
Check this for dubious points as $x=1$ or any other that you dont see clearly if belong or not to some set.
A: How to formally solve:
Well, To be thoruogh. 
If $x \le -1$ then $x \not \in (-1, 1+ 1/n)$ for any $n$ so $x \not \in \cap (-1, 1+ 1/n)$.
If $-1 < x \le 1$ then $-1 < x \le 1 < 1 + 1/n$ for all $n$ so $x \in \cap (-1, 1+ 1/n)$.
if $x > 1$ then $x - 1 > 0$ so there exists a natural number $n$ so that $x-1 > 1/n > 0$.  So $x \not \in (-1, 1+1/n)$ so $x \not \in \cap (-1, 1+ 1/n)$.
So $x \in \cap (-1, 1+ 1/n) \iff -1 < x \le 1$ so $\cap (-1, 1+ 1/n) = (-1, 1]$.
The second part similarly.  if $0 \le x < 1$ then $1 -x > 0$ so there exists a $n$ so the $1 - x > 1/n > 0$ so $0\le x < 1 - 1/n$ so $x \in [0, 1-1/n]$ so $x \in \cup [0,1-1/n]$ but if $x \ge 1$ then $x \ge 1 > 1-1/n$ for all $n$ so $x$ is not in union and obviously if $x < 0$ then $x$ is not in the union.
