Should $\bigcap_{n = 1}^\infty (a-\frac{1}{n}, b + \frac{1}{n})$ be $(a,b)$ or $[a,b]$ I am confused about the limiting behavior of as $n \to \infty$, $\bigcap_{n = 1}^\infty (a-\frac{1}{n}, b + \frac{1}{n})$. 
I have read that it is the case that this set becomes closed, but I can't help but to think that if we take $n \to \infty$, then $\dfrac{1}{n} \to 0$, and the inner set is $(a,b)$
Can someone please help?
 A: You say you're concerned about the limiting behavior. You talk about the set "becoming" closed, and about letting $n\to\infty$. All of that shows you're thinking about this totally wrong.
It's important to realize here that the notation $$A=\bigcap_{n=1}^\infty A_n$$has nothing to do with limits! It means that $A$ is the intersection of all the sets $A_n$.  Not the limit of the intersection of the first $N$, it's the intersection of all of them, period.
Like in calculus an infinite sum is not really the sum of infinitely many numbers, because we can't add infinitely many numbers - an infinite sum is really the limit of finite sums? Understanding that is a good thing, but that's leading you astray here. This is not like that at all - that intersection is the intersection of infinitely many sets.
What the notation does mean is that $x\in A$ if and only if $x\in A_n$ for every $n$.
So go back to your problem. You should be able to convince yourself that given $x$, you have $x\in(a-1/n,b+1/n)$ for every $n$ if and only if $x\in[a,b]$. And there you  are.
