Question

Construct a bounded set of real numbers with exactly three limit points.

Short answer

Question: Construct a bounded set of real numbers that has exactly three limit points and provide an example. Answer: One example of a bounded set of real numbers with exactly three limit points is the set \(S = \{-1 + \frac{1}{n} , \frac{1}{2n}, 1 - \frac{1}{n} \ | \ n \in \mathbb{N}\}\). This set has limit points at -1, 0, and 1, and is bounded.

Step-by-step solution

Choose three distinct real numbers

Let's choose three different real numbers that will act as our limit points. For instance, we can take ,-1, 0, and 1.

Construct sequences that converge to limit points

For each of our chosen real numbers, we will construct a sequence that converges to that number. Let's consider the following sequences: - For -1: \(a_n = -1 + \frac{1}{n}\) - For 0: \(b_n = \frac{1}{2n}\) - For 1: \(c_n = 1 - \frac{1}{n}\) Notice how each sequence converges to the corresponding chosen real number and how all sequences are distinct.

Combine the sequences into a single set

Since we have established the sequences that converge to our limit points, now let's combine those sequences into a single set. This set will be the desired bounded set of real numbers with three limit points. Denote S as this set: \(S = \{a_n\} \cup \{b_n\} \cup \{c_n\}\) \(S = \{-1 + \frac{1}{n} , \frac{1}{2n}, 1 - \frac{1}{n} \ | \ n \in \mathbb{N}\}\)

Confirm that the original conditions are satisfied

We must ensure that our set S has exactly three limit points and is bounded to validate our construction. First, let's verify that the limit points are -1, 0, and 1. By definition, a limit point of a set is a point such that any open interval around it contains points of the set, other than itself. In our set S, since the sequences \(a_n\), \(b_n\), and \(c_n\) converge to -1, 0, and 1 respectively, in any open interval around -1, 0, and 1, we can find points that come from these sequences, thus making them limit points. Now, we must show that there are no other limit points. Notice that the sequences and set S were constructed in a way that every element of the set is in the vicinity of one of the predefined limit points. Having a fourth limit point outside this vicinity would require elements approaching it, which are absent from S. Therefore, S has only three limit points, -1, 0, and 1. Lastly, we show that our set S is bounded. The lower bound for the set is -2 and the upper bound is 2. Indeed, no element in our set S goes below -2 or above 2. For any n, \(-2 < -1 + \frac{1}{n} \le 0\), \(0 < \frac{1}{2n} \le 1\), and \(0 \le 1 - \frac{1}{n} < 2\). Thus, our constructed set S is bounded. Our desired set of real numbers is \(S = \{-1 + \frac{1}{n} , \frac{1}{2n}, 1 - \frac{1}{n} \ | \ n \in \mathbb{N}\}\), which has exactly three limit points and is bounded.

Key concepts

Limit Points

In mathematics, a limit point, also known as an accumulation point, is a crucial concept in the study of sets and sequences. Imagine a point in a set where you can always find other elements of the set, no matter how close you zoom in.
For a point to be a limit point of a set, every open interval around that point must contain at least one other different point from the set. For example, with our constructed set of real numbers, the numbers -1, 0, and 1 serve as limit points.
Each of them fulfills the condition:

• Any small neighborhood around -1 contains elements from the sequence \(a_n = -1 + \frac{1}{n}\).
• Any open interval around 0 contains points from \(b_n = \frac{1}{2n}\).
• Any open interval around 1 contains points from \(c_n = 1 - \frac{1}{n}\).

Recognizing limit points helps in understanding the structure and boundaries of sets in real analysis.

Real Numbers

Real numbers encompass a broad category of numbers, including integers, fractions, and irrational numbers, which form a line on the number line. They are essential in everyday math and calculus, providing a framework for measuring continuous quantities.
In our exercise, we've constructed a set using real numbers where the sequences have specific limit points (-1, 0, and 1). Real numbers are excellent here because they allow for infinite divisibility. This property lets us create sequences like \(a_n = -1 + \frac{1}{n}\) that converge closer and closer to -1.
Working with real numbers is vital for modeling real-world situations and solving equations that involve both rational and irrational values. From graphs to advanced calculations, their importance cannot be overstated.

Mathematical Sequences

A sequence in mathematics is simply a list of numbers in a specific order, often defined by a formula. Understanding sequences is a stepping stone to grasping more advanced topics like series and calculus.
In our exercise, sequences \(a_n = -1 + \frac{1}{n}\), \(b_n = \frac{1}{2n}\), and \(c_n = 1 - \frac{1}{n}\) are used strategically. Each sequence converges to a particular limit point (-1, 0, or 1 respectively).
Here's what makes sequences so interesting:

• They help describe behavior as numbers progress towards infinity.
• By using rules or formulas, we can predict and find patterns in data.
• Convergence, where sequences get closer to a specific value, is key in calculus and real analysis.

Whether it's modeling growth, examining limits, or solving complex equations, sequences are a fundamental concept that aids in the understanding of continuous functions.