Chapter 7.1, Problem 3E

(a)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $\{\frac{1}{n}:n\in \mathbb{N}\}$ .

The set that is given is $\{\frac{1}{n}:n\in \mathbb{N}\}$ . We can see that the set is bounded both up and below. The minimum upper bound is 1 and the largest lower bound is 0. Therefore, the infimum will

be 0 and the supremum will be 1.

Hence, both infimum and supremum exist.

(b)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $\{\frac{(n+1)}{n}:n\in \mathbb{N}\}$ .

The set that is given is $\{\frac{(n+1)}{n}:n\in \mathbb{N}\}$ . We can see that the set is bounded both up and below.

The given set is decreasing, $2\ge \frac{(n+1)}{n}=1+\frac{1}{n}$ for all possible values of $n\in \mathbb{N}$ . Therefore, the largest upper bound is 2 and the lowest lower bound is 1. Therefore, the infimum will

be 1 and the supremum will be 2.

Hence, both infimum and supremum exist.

(c)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Infimum exists and supremum doesn’t exist.

Explanation of Solution

Given Information:

The given set is $\{2^x:x\in \mathbb{Z}\}$ .

The set that is given is $\{2^x:x\in \mathbb{Z}\}$ . We can see that $2^x>0$ for all possible values of $x\in \mathbb{Z}$ .

Therefore, the set has no upper bound and the lower bound is 0. Therefore, the infimum will

be 0 and no supremum exists.

Hence, infimum exists and supremum doesn’t exist.

(d)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $\{{(-1)}^n(1+\frac{1}{n}):n\in \mathbb{N}\}$ .

The set that is given is $\{{(-1)}^n(1+\frac{1}{n}):n\in \mathbb{N}\}$ . We can see that $-(1+\frac{1}{n})\le {(-1)}^n(1+\frac{1}{n})\le (1+\frac{1}{2})$ Therefore, the infimum will be $-2$ and the supremum will be $\frac{3}{2}$ .

Hence, both infimum and supremum exist.

(e)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $\{\frac{n}{n+2}:n\in \mathbb{N}\}$ .

The set that is given is $\{\frac{n}{n+2}:n\in \mathbb{N}\}$ . We can see that the set is an increasing set with the condition of $\frac{1}{3}\le \frac{n}{n+2}<1$ for all the possible values of $n\in \mathbb{N}$ . Therefore, the infimum will be $\frac{1}{3}$ and supremum will be $1$ .

Hence, both infimum and supremum exist.

(f)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $\{x\in \mathbb{Q}:x^2<10\}$ .

The set that is given is $\{x\in \mathbb{Q}:x^2<10\}$ . We can see that the value of $x$ is between $-\sqrt{10}<x<\sqrt{10}$ . Therefore, the infimum will be $-\sqrt{10}$ and supremum will be $\sqrt{10}$ .

Hence, both infimum and supremum exists.

(g)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $[-1,1]\cup \left\{5\right\}$ .

The set that is given is $[-1,1]\cup \left\{5\right\}$ . We can see that the value of $x$ is between $-1\le x\le 5$ . Therefore, the infimum will be $-1$ and supremum will be $5$ .

Hence, both infimum and supremum exist.

(h)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum exist.

Explanation of Solution

Given Information:

The given set is $[-1,1]-\left\{0\right\}$ .

The set that is given is $[-1,1]-\left\{0\right\}$ . We can see that all the values of the set are between $-1$ and $1$ . Therefore, the infimum will be $-1$ and supremum will be $1$ .

Hence, both infimum and supremum exist.

(i)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Infimum exist but supremum doesn’t exist.

Explanation of Solution

Given Information:

The given set is $\{2^{-y}x:x,y\in \mathbb{N}\}$ .

The set that is given is $\{2^{-y}x:x,y\in \mathbb{N}\}$ . We can see that for all the large values of $y$ , $2^{-y}x\to 0$ . Therefore, the infimum will be $0$ and no supremum exists as for a fixed $y,x{2}^{-y}$ keeps increasing.

Hence, infimum exist but supremum doesn’t exist.

(j)

To determine

To find: The supremum and infimum of the given set.

Answer to Problem 3E

Both infimum and supremum doesn’t exist.

Explanation of Solution

Given Information:

The given set is $\{x:|x|>2\}$ .

The set that is given is $\{x:|x|>2\}$ . We can see that the given set has the criteria of $(-\infty ,-2)\cup (2,\infty )$ . Therefore, infimum and supremum doesn’t exist.

Hence, both infimum and supremum doesn’t exist.