Question

(3 Points) Prove ⋃n∈N​[n1​,1]=(0,1] Hint: use the ceiling function introduced in HW5 Problem 2 (and probably you will use Question 2 of that Problem).

Answer 1

The question requires proof that the union of intervals [1/n, 1] for all natural numbers n equals (0,1]. This is done using the ceiling function to establish that any x in (0,1] can be associated with an interval [1/n, 1], therefore concluding the union of such intervals equals (0,1].

Step-by-step explanation:

The student's question asks to prove that the union of the intervals [1/n, 1] for all n in the natural numbers is equal to the interval (0,1]. This can be proved using the ceiling function. For any x in (0,1], we can find a natural number n such that 1/n < x, which implies x is in the interval [1/n,1]. Conversely, any x in the union of these intervals must be in the interval (0,1] since x cannot be 0 (as n is natural and so 1/n is always positive) and x ≤ 1 by definition of the intervals. Therefore, ∪n∈N [1/n, 1] = (0,1].