finding intersection and union of indexed family of subsets of real number 
I'm guessing that the intersection would be empty set and union would be set of real number. How would I be able to prove it, using Archimedean Property if necessary? 

You need to look at the definition of intersection and union and apply them directly. When in doubt, always look at the relevant definition(s).
Intersection:

$$\require{color}\bigcap_{i \in I} A_i = \{x : x \in A_i {\color{blue}\text{ for all }} i\} $$

Union:

$$ \bigcup_{i \in I} A_i = \{x : x \in A_i {\color{blue}\text{ for some }} i\} $$

What does it mean for $x \in A_t$ for all $t$? It means $x \ge t$ for all $t$ because "$x \in A_t$" is equivalent to "$x \ge t$." Likewise, $x \in A_t$ for some $t$ means $x \ge t$ for some $t$.
