Question

1. Suppose ?:? → ? and {??}?∈? is an indexed collection of subsets of set ?. Prove ?(⋂ ?? ?∈? ) ⊂ ⋂ ?(??) ?∈? with equality if ? is one-to-one.

2. Compute:

a. ⋂ ∞ ?=1 [?,∞)

b. ⋃ ∞ ?=1 [0,2 − 1 /?]

c. lim sup ?→∞ (−1 + (−1)^? /?,1 +(−1)^? /?)
d. lim inf ?→∞(−1 +(−1)^?/ ?,1 +(−1)^? /?)

Answer 1

1. Let , then there exists , such that .

Now for all , this implies for all , this implies . Thus we get  .

Now suppose assume f is 1-1. Then to show equality we take , for all . This gives us for all there exists , such that , Now since f is 1-1 and  , for any , gives us (say). Thus we get a unique , for all (as ), such that . Now this gives us , hence we get . Hence the equality.

2.

(i)  , as if there exists , then we can find such that this gives us , which is a contradiction hence our claim is established.

(ii) .

Note that clearly for all , we have .

Now suppose take ,then we can find , be such that, (as this follows from the fact and ) this gives us , Hence the equality.

(iii)

(iv)