Question

Show that the following sequences of sets, \(\left\\{C_{k}\right\\}\), are nonincreasing, \((1.2 .17)\), then find \(\lim _{k \rightarrow \infty} C_{k}\). (a) \(C_{k}=\\{x: 2-1 / k

Short answer

All three sequences of sets are nonincreasing. The limits as \(k\) goes to infinity are: For the first sequence, \{2\}, for the second sequence, the empty set \(\emptyset\) and for the third sequence, the set \{(0,0)\}.

Step-by-step solution

Unpack the definition of nonincreasing

A sequence of sets \(\{C_{k}\}\) is said to be nonincreasing if and only if for every \(k\), \(C_{k} \supseteq C_{k+1}\), i.e., every set in the sequence contains the next one.

Show nonincreasing for first sequence

For set \(C_{k} = \{x: 2-1/k<x \leq 2\} , k=1,2,3,\ldots\), as \(k\) increases, the left bound of the interval increases and moves towards 2. Thus every subsequent set becomes a subset of the previous set, hence, the sequence is nonincreasing.

Show nonincreasing for second sequence

For set \(C_{k} = \{x: 2<x \leq 2+1/k\} , k=1,2,3,\ldots\), as \(k\) increases, the upper bound of the interval decreases and moves towards 2. Thus every subsequent set becomes a subset of the previous set, hence, the sequence is nonincreasing.

Show nonincreasing for third sequence

For set \(C_{k} = \{(x, y): 0 \leq x^{2}+y^{2} \leq 1/k\} , k=1,2,3,\ldots\), as \(k\) increases, the radius of the disk decreases and its area becomes smaller. Hence, every subsequent set becomes a subset of the previous set and the sequence is nonincreasing.

Find limit of the first sequence

The limit of \(C_{k} = \{x: 2-1/k<x \leq 2\}, k=1,2,3,\ldots\) as \(k\) goes to infinity is given by \(\lim_{k \rightarrow \infty}C_{k} = \{x: 2 \leq x \leq 2\} = \{2\}\). This is because as \(k\) increases, the left bound of the interval moves towards 2.

Find limit of the second sequence

The limit of \(C_{k} = \{x: 2<x \leq 2+1/k\} , k=1,2,3,\ldots\) as \(k\) goes to infinity is \(\lim_{k \rightarrow \infty}C_{k} = \{x: 2 < x \leq 2\} = \emptyset\). As \(k\) increases, the right bound of the interval moves towards 2.

Find limit of the third sequence

The limit of \(C_{k} = \{(x, y): 0 \leq x^{2}+y^{2} \leq 1/k\}, k=1,2,3,\ldots\) as \(k\) goes to infinity is given by \(\lim_{k \rightarrow \infty}C_{k} = \{(x, y): 0 \leq x^{2}+y^{2} \leq 0\} = \{(0,0)\}\). This is because as \(k\) increases, the radius of the disk goes to zero.