Question

Find the supremum and infimum of each \(S .\) State whether they are in \(S\). (a) \(S=\left\\{x \mid x=-(1 / n)+\left[1+(-1)^{n}\right] n^{2}, n \geq 1\right\\}\) (b) \(S=\left\\{x \mid x^{2}<9\right\\}\) (c) \(S=\left\\{x \mid x^{2} \leq 7\right\\}\) (d) \(S=\\{x|| 2 x+1 \mid<5\\}\) (e) \(S=\left\\{x \mid\left(x^{2}+1\right)^{-1}>\frac{1}{2}\right\\}\) (f) \(S=\left\\{x \mid x=\right.\) rational and \(\left.x^{2} \leq 7\right\\}\)

Short answer

Question: Determine the supremum and infimum of each set, if they exist, and state whether the supremum and infimum are in the set. (a) \(S=\left\\{x \mid x=-(1 / n)+\left[1+(-1)^{n}\right] n^{2}, n \geq 1\right\\}\) Answer: The infimum is 0 (and it belongs to S), and the supremum is \(\infty\) (but it does not belong to S). (b) \(S=\left\\{x \mid x^{2}<9\right\\}\) Answer: The infimum is -3 (but it does not belong to S), and the supremum is 3 (but it does not belong to S). (c) \(S=\left\\{x \mid x^{2} \leq 7\right\\}\) Answer: The infimum is \(-\sqrt{7}\) (and it belongs to S), and the supremum is \(\sqrt{7}\) (and it belongs to S). (d) \(S=\{x\mid 2x+1 < 5\}\) Answer: The infimum is \(-\infty\) (but it does not belong to S), and the supremum is 2 (but it does not belong to S). (e) \(S=\{x\mid (x^{2}+1)^{-1}>\frac{1}{2}\}\) Answer: The infimum is -1 (but it does not belong to S), and the supremum is 1 (but it does not belong to S). (f) \(S=\{ x \mid x \textrm{ is rational and } x^{2} \leq 7\}\) Answer: The infimum is \(-\sqrt{7}\) (but it does not belong to S because it is irrational), and the supremum is \(\sqrt{7}\) (but it does not belong to S because it is irrational).

Step-by-step solution

(a) Analyzing the set S

We need to find the supremum and infimum of the set \(S=\left\\{x \mid x=-(1 / n)+\left[1+(-1)^{n}\right] n^{2}, n \geq 1\right\\}\). We can first simplify the expression inside the set, and then find out the bounds of the set S, which will help us find the supremum and infimum.

(a) Simplifying the expression

We can notice that when n is odd, \((-1)^{n}=-1\), and when n is even, \((-1)^{n}=1\). So, we can split the set into two sequences, one for even values of n and one for odd values of n.

(a) Finding the bounds of the set S

By splitting the sequence, we can see that: For odd n: \(x_{odd}=-(1 /n)+ [1+ (-1)]n^{2}=-(1 / n)\) For even n: \(x_{even}=-(1 /n)+ [1+ 1] n^{2}=n^{2}-(1/n)\) Now we can find the bounds for each sequence: For Odd n: \(\lim_{n\to\infty} -\frac{1}{n} = 0\) For Even n: \(\lim_{n\to\infty}(n^{2}-\frac{1}{n})=\infty\)

(a) Finding supremum and infimum and stating whether they are in S

From our bounds analysis, we can see that the infimum is 0 (and it belongs to S) and the supremum is \(\infty\) (but it does not belong to S).

(b) Analyzing the set S

For the set \(S=\left\\{x \mid x^{2}<9\right\\}\), we need to find the supremum and infimum.

(b) Finding the bounds of the set S

Here, the values of x satisfy the inequality \(x^2 < 9\). We can solve this inequality as follows: \(-3 < x < 3\).

(b) Finding supremum and infimum and stating whether they are in S

From the bounds, we can see that the infimum is -3 (but it does not belong to S) and the supremum is 3 (but it does not belong to S).

(c) Analyzing the set S

For the set \(S=\left\\{x \mid x^{2} \leq 7\right\\}\), we need to find the supremum and infimum.

(c) Finding the bounds of the set S

Here, the values of x satisfy the inequality \(x^2 \leq 7\). We can solve this inequality as follows: \(-\sqrt{7} \leq x \leq \sqrt{7}\)

(c) Finding supremum and infimum and stating whether they are in S

From the bounds, we can see that the infimum is \(-\sqrt{7}\) (and it belongs to S) and the supremum is \(\sqrt{7}\) (and it belongs to S).

(d) Analyzing the set S

For the set \(S=\{x\mid 2x+1 < 5\}\), we need to find the supremum and infimum.

(d) Finding the bounds of the set S

Here, the values of x satisfy the inequality \(2x+1 < 5\). We can solve this inequality as follows: \(x < 2\).

(d) Finding supremum and infimum and stating whether they are in S

From the bounds, we can see that the infimum is \(-\infty\) (but it does not belong to S) and the supremum is 2 (but it does not belong to S).

(e) Analyzing the set S

For the set \(S=\{x\mid (x^{2}+1)^{-1}>\frac{1}{2}\}\), we need to find the supremum and infimum.

(e) Finding the bounds of the set S

Here, we can solve the inequality as follows: \(|\frac{1}{x^2 + 1}| > \frac{1}{2}\), which simplifies to \(x^2 < 1\), so \(-1 < x < 1\).

(e) Finding supremum and infimum and stating whether they are in S

From the bounds, we can see that the infimum is -1 (but it does not belong to S) and the supremum is 1 (but it does not belong to S).

(f) Analyzing the set S

For the set \(S=\{ x \mid x \textrm{ is rational and } x^{2} \leq 7\}\), we need to find the supremum and infimum.

(f) Finding the bounds of the set S

The inequality \(x^2 \leq 7\) is satisfied for rationals x such that \(-\sqrt{7} \leq x \leq \sqrt{7}\).

(f) Finding supremum and infimum and stating whether they are in S

From the bounds, we can see that the infimum is \(-\sqrt{7}\) (but it does not belong to S because it is irrational) and the supremum is \(\sqrt{7}\) (but it does not belong to S because it is irrational).

Key concepts

Bounded Sets

In real analysis, a set is considered bounded when its elements lie within a certain fixed distance from a reference point, usually the origin. Specifically, a set is bounded from above if there exists a real number that is greater than or equal to every element in the set. Similarly, a set is bounded from below if there is a real number less than or equal to every element. If a set is both bounded from above and below, it is simply bounded.

For example, in the exercise provided, the set S in part (b) containing all numbers x such that x^2 < 9 is bounded, as the value of x lies within the interval (-3, 3). However, we should note that in this case, the numbers -3 and 3 are not part of S, given the strict inequality. Therefore, -3 and 3 act as the bounds but are not the supremum or infimum of S themselves. Understanding boundedness is important for analyzing the behavior of sequences and functions, and for determining the existence of supremums (least upper bounds) and infimums (greatest lower bounds).

Inequalities in Analysis

In real analysis, inequalities are used to express the relationships between elements within a set or between different sets. These relationships help establish the existence of supremums and infimums. A supremum is the smallest number that is greater than or equal to every element in the set, while the infimum is the largest number that is less than or equal to every element in the set.

In the exercise (c), where the set S contains all x such that x^2 ≤ 7, the inequalities help us determine that the values of x are confined between -√7 and √7. This knowledge allows us to conclude that √7 is the supremum and -√7 is the infimum, and both are part of S since the inequality is non-strict (i.e., using ≤ instead of <). Inequalities can sometimes tell us more than whether a value is simply in or out of a set—they can define intervals for the sets and are crucial in the solution of many analytical problems.

Limits of Sequences

A fundamental aspect of real analysis is understanding the limit of a sequence, which is the value that the elements of a sequence 'approach' as the index (usually denoted as n) goes to infinity. If a sequence has a limit, we often investigate its behavior in relation to finding supremums and infimums of related sets.

In part (a) of the exercise, for even n, the sequence x_even equals n^2 - (1/n). As n approaches infinity, the 1/n term vanishes, and the sequence grows without bound—heading to ∞. On the other hand, the sequence for odd n, given by x_odd = -(1/n), approaches 0 as n becomes very large. This demonstrates that the supremum of the set S in this example is infinity (since part of the sequence grows without bound), and the infimum is 0 (as that is where the other part of the sequence converges), with 0 being an element of S. Recognizing and calculating the limits of sequences is essential for a comprehensive understanding of their behavior and for the correct determination of bounds within sets in analysis.