Show that the following sequences of sets, {Ck}, are non-decreasing or non-increasing give definition of limit k→∞ Ck and then find it in each of the following cases.

Question

Show that the following sequences of sets, {C_k}, are non-decreasing or non-increasing

give definition of limit k→∞ C_k and then find it in each of the following cases.

(a) {Ck} = {x : x ≥ 9 - (1/k)} , k= 1, 2, ...

(b) {Ck} = { (x,y) : 1 + (1/k) ≤ x² +y²}, k = 1, 2, 3, ...

(c) {Ck} = {x : 0 < x < 1 - (1/k)}, k = 1, 2, 3, ...

Answer 1

a) 9-1/k <9 - 1/(k+1) which implies C(k+1) is a subset of Ck. So Ck is decreasing. The limit is [9, +infinity)

b) 1+1/(k+1) <1 + 1/k which implies that Ck is a subset of C(k+1). So Ck is increasing. The limit is outside the circle x^2 + y^2 = 1, that is {(x, y) | x^2 + y^2 >=1}

c) 1-1/k < 1-1/(k+1) which implies that Ck is a subset of C(k+1). So Ck is increasing. The limit is (0, 1).

In each case 1/k tends to 0 as k goes to infinity.