Question

Complete all parts of the problem.

Answer 1

(a)

$\bigcup\limits_{i=1}^7R_i=R_1=\boxed{[1,2]}$

(b)

$\bigcap\limits_{i=1}^7R_i=R_7=\boxed{\left[1,\dfrac87\right]}$

(c) No, because no two sets are disjoint. Why? Each of the $R_i$ contain the endpoint 1, so at the very least, $\{1\}\subseteq R_i\cup R_j$ for $i\neq j$.

(d)

$\bigcup\limits_{i=1}^nR_i=R_1=\boxed{[1,2]}$

(e)

$\bigcap\limits_{i=1}^nR_i=R_n=\boxed{\left[1,\dfrac{n+1}n\right]}$

(f)

$\bigcup\limits_{i=1}^\infty R_i=R_1=\boxed{[1,2]}$

because as i gets larger, the set $R_i$ gets smaller. The infinite union will be equivalent to the largest set in the family of sets.

(g)

$\bigcap\limits_{i=1}^\infty R_i=\boxed{\{1\}}$

because 1 + 1/n converges to 1 as n goes to infinity, so $R_i$ converges to the singleton set {1} as i goes to infinity.