How do you simplify in writing $[1, \infty) \cap [2, \infty) \cap [3, \infty) \ldots $ , in Real Analysis? I know the answer to this very basic question in plain language but I want to know the right way to say it in real analysis:
$$\begin{align}
\text{If} \ A_t &= \{x \mid t \leq x\}, \\
\bigcap_{t \in \mathbb N}A_t &= [1, \infty) \cap [2, \infty) \cap [3, \infty) \ldots \\
&= \ldots
\end{align}$$
Which one should be the correct answer here?
(1) $[n, \infty)$, where $n = \max \ \mathbb N$ 
(2) $[n, \infty)$, where $n = \sup \ \mathbb N$ 
(3) $(\infty, \infty)$.
Thank you very much for your time. 
 A: Actually, none of those options are correct,  since $\Bbb N$ is unbounded neither $\max\Bbb N$ nor $\sup\Bbb N$ exist.  In fact,
$$\bigcap_{n\in\Bbb N}[n,\infty)=\emptyset.$$
The notation $(\infty,\infty)$ can be considered correct I suppose, but $(\infty,\infty)=\emptyset$.
A: The correct answer is $\emptyset$. (1) is actually wrong, since $\max{\mathbb{N}}$ doesn't exist. The other two are technically correct but suggest a lack of understanding of $\infty$ (it's not a number and you can't treat it like one!)
A: First of all, $\{\mathbb{N}\}$ is a set with one element, and that element is a set; in particular, $\{\mathbb{N}\} \not\subseteq \mathbb{R}$, so $\max\{\mathbb{N}\}$ and $\sup\{\mathbb{N}\}$ don't make sense. I believe you mean $\operatorname{max}\mathbb{N}$ and $\sup\mathbb{N}$.
As $\mathbb{N}$ is not bounded above, both $\operatorname{max}\mathbb{N}$ and $\sup\mathbb{N}$ don't exist, so $(1)$ and $(2)$ are incorrect. As for $(3)$, I don't know what $(\infty, \infty)$ is supposed to mean; usually such an interval isn't defined. Intuitively, it should be the set with no elements, i.e. the empty set $\emptyset$. The empty set is the correct answer because if $y \in \bigcap_{t \in \mathbb{N}}A_t$, then $y \in A_t$ for every $t \in \mathbb{N}$, so $y \geq t$ for every $t \in \mathbb{N}$, but no such element exists.
A: The sets $A_t$ in this question are tacitly supposed to be sets of real numbers. Since by the Archimedian principle for every real number $x$ there is a natural number $n>x$  the intersection $\bigcap_{\>t\in{\mathbb N}} A_t$ is empty. Everything else is bullshit and trying to make a measly sense out of a poorly composed exercise.
