Proving The Limit Of A Product Of Sequences A Step-by-Step Guide

In the realm of mathematical analysis, understanding the behavior of sequences and their limits is fundamental. One common scenario involves determining the limit of a sequence formed by the product of two other sequences. Specifically, if we have two sequences, $(x_n)$ and $(y_n)$, that converge to limits $x$ and $y$, respectively, a natural question arises: Does the sequence $(x_n y_n)$ converge to the product $xy$? This article delves into a rigorous proof of this result, providing a step-by-step explanation and highlighting key concepts along the way. We will explore the formal definition of a limit, the properties of convergent sequences, and how these elements combine to establish the desired conclusion.

1. Introduction to Sequence Convergence

Before diving into the proof, let's refresh our understanding of sequence convergence. A sequence $(x_n)$ is said to converge to a limit $x$ if, for any arbitrarily small positive number $\epsilon$, there exists a positive integer $N$ such that the distance between $x_n$ and $x$ is less than $\epsilon$ for all $n$ greater than $N$. Formally, this is expressed as:

$\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } |x_n - x| < \epsilon \text{ for all } n > N.$

This definition captures the intuitive idea that as $n$ becomes larger, the terms of the sequence get arbitrarily close to the limit $x$. The choice of $N$ depends on $\epsilon$, reflecting the fact that we can make the terms of the sequence as close to $x$ as we desire by going far enough along in the sequence. Understanding this definition is crucial for comprehending the subsequent proof.

In the context of our problem, we are given that $\lim_{n \to \infty} x_n = x$ and $\lim_{n \to \infty} y_n = y$. This means that for any given $\epsilon_1 > 0$, we can find an integer $N_1$ such that $|x_n - x| < \epsilon_1$ for all $n > N_1$. Similarly, for any given $\epsilon_2 > 0$, we can find an integer $N_2$ such that $|y_n - y| < \epsilon_2$ for all $n > N_2$. These inequalities will form the foundation of our proof.

2. Key Properties of Convergent Sequences

To prove that $\lim_{n \to \infty} (x_n y_n) = xy$, we will leverage several important properties of convergent sequences. One such property is that convergent sequences are bounded. This means that if a sequence $(x_n)$ converges, there exists a positive number $M$ such that $|x_n| \leq M$ for all $n$. This property is crucial because it allows us to control the magnitude of the terms in the sequence. The boundedness property arises from the fact that after a certain point in the sequence (beyond some index $N$), the terms are close to the limit, and before that point, there are only finitely many terms, each of which has a finite magnitude.

Another crucial property is the algebra of limits. This property states that if $\lim_{n \to \infty} x_n = x$ and $\lim_{n \to \infty} y_n = y$, then:

• $\lim_{n \to \infty} (x_n + y_n) = x + y$
• $\lim_{n \to \infty} (c x_n) = c x$ for any constant $c$
• $\lim_{n \to \infty} (x_n y_n) = xy$

Our goal is to prove the third part of the algebra of limits, which concerns the product of sequences. While the algebra of limits provides a powerful tool for manipulating limits, we will provide a direct proof from the definition of a limit to gain a deeper understanding of the underlying principles.

In addition to boundedness, the algebra of limits related to sums and scalar products will indirectly be used within our proof as we manipulate inequalities and construct our argument. For instance, we'll use the triangle inequality, which is closely related to the sum of sequences property.

3. Proof of the Limit of a Product

Now, let's embark on the proof that if $\lim_{n \to \infty} x_n = x$ and $\lim_{n \to \infty} y_n = y$, then $\lim_{n \to \infty} (x_n y_n) = xy$. The core idea is to show that for any given $\epsilon > 0$, we can find an integer $N$ such that $|x_n y_n - xy| < \epsilon$ for all $n > N$. To achieve this, we will manipulate the expression $|x_n y_n - xy|$ and use the convergence of $(x_n)$ and $(y_n)$ to bound it by $\epsilon$.

We begin by adding and subtracting $x_n y$ inside the absolute value, a common technique used to bridge the gap between $x_n y_n$ and $xy$:

$|x_n y_n - xy| = |x_n y_n - x_n y + x_n y - xy|$

Next, we apply the triangle inequality, which states that $|a + b| \leq |a| + |b|$ for any real numbers $a$ and $b$:

$|x_n y_n - x_n y + x_n y - xy| \leq |x_n y_n - x_n y| + |x_n y - xy|$

We can factor out common terms in each absolute value:

$|x_n y_n - x_n y| + |x_n y - xy| = |x_n| |y_n - y| + |y| |x_n - x|$

Now, we leverage the fact that convergent sequences are bounded. Since $(x_n)$ converges to $x$, there exists a positive number $M$ such that $|x_n| \leq M$ for all $n$. This allows us to bound the first term:

$|x_n| |y_n - y| + |y| |x_n - x| \leq M |y_n - y| + |y| |x_n - x|$

Here's where the convergence of $(x_n)$ and $(y_n)$ comes into play. We know that for any $\epsilon_1 > 0$, there exists $N_1$ such that $|x_n - x| < \epsilon_1$ for all $n > N_1$. Similarly, for any $\epsilon_2 > 0$, there exists $N_2$ such that $|y_n - y| < \epsilon_2$ for all $n > N_2$. Let's choose $\epsilon_1 = \frac{\epsilon}{2|y| + 1}$ and $\epsilon_2 = \frac{\epsilon}{2M}$. Note that we add 1 to $|2y|$ in the denominator of $\epsilon_1$ to avoid division by zero if $y = 0$.

Substituting these choices into the inequality, we get:

$M |y_n - y| + |y| |x_n - x| < M \frac{\epsilon}{2M} + |y| \frac{\epsilon}{2|y| + 1} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$

Let $N = \max(N_1, N_2)$. Then for all $n > N$, both $|x_n - x| < \frac{\epsilon}{2|y| + 1}$ and $|y_n - y| < \frac{\epsilon}{2M}$ hold. Therefore, we have shown that for any $\epsilon > 0$, there exists an integer $N$ such that $|x_n y_n - xy| < \epsilon$ for all $n > N$. This is precisely the definition of $\lim_{n \to \infty} (x_n y_n) = xy$, thus completing the proof.

4. Implications and Applications

The result we have proven, that the limit of the product of two convergent sequences is the product of their limits, has significant implications in various areas of mathematics. It allows us to compute limits of more complex expressions involving sequences. For instance, if we have a sequence defined as a product of simpler sequences, we can find its limit by finding the limits of the individual sequences and then multiplying them.

This result is also crucial in the development of calculus and real analysis. It is used in proving other limit theorems and in defining concepts such as continuity and differentiability. When dealing with functions, we often consider sequences of function values, and the limit of a sequence of products of functions can be determined using this principle.

Moreover, this theorem provides a foundation for numerical analysis, where sequences are used to approximate solutions to equations or values of functions. By understanding how the limits of sequences behave under multiplication, we can develop more accurate and efficient numerical methods.

In summary, the proof that $\lim_{n \to \infty} (x_n y_n) = xy$ is not just a theoretical exercise; it is a fundamental result with wide-ranging applications in mathematics and related fields. Its understanding provides a deeper appreciation of the nature of limits and their role in mathematical analysis.

5. Common Pitfalls and Misconceptions

When working with limits of sequences, several common pitfalls and misconceptions can arise. One frequent mistake is assuming that if the terms of two sequences get closer to each other, their limits must be equal. While this might seem intuitive, it is not always the case. For example, consider the sequences $x_n = \frac{1}{n}$ and $y_n = \frac{1}{n^2}$. As $n$ approaches infinity, the terms $x_n$ and $y_n$ both approach 0, and their difference also approaches 0. However, the sequence $\frac{x_n}{y_n} = n$ diverges to infinity, demonstrating that the ratio of two sequences converging to 0 can behave in various ways.

Another common misconception is that the limit of a sequence always exists. Sequences can diverge in several ways: they can oscillate, tend to infinity, or not settle down to any particular value. It is crucial to verify that a limit exists before attempting to compute it. In our proof, we relied on the assumption that both $\lim_{n \to \infty} x_n$ and $\lim_{n \to \infty} y_n$ exist; without this assumption, the proof would not hold.

Furthermore, students sometimes struggle with the ${\epsilon-N}$ definition of a limit. The concept of choosing $N$ based on $\epsilon$ can be challenging to grasp. It is important to remember that ${\epsilon}$ represents an arbitrarily small tolerance, and we need to show that we can make the terms of the sequence fall within this tolerance by going far enough along in the sequence (i.e., choosing a sufficiently large $N$).

In the context of our proof, a common mistake is not properly bounding the terms in the expression $|x_n y_n - xy|$. The use of the triangle inequality and the boundedness of convergent sequences are crucial steps that must be applied correctly to arrive at the desired result. Choosing appropriate values for $\epsilon_1$ and $\epsilon_2$ is also essential to ensure that the final inequality holds.

6. Conclusion

In conclusion, we have rigorously proven that if $\lim_{n \to \infty} x_n = x$ and $\lim_{n \to \infty} y_n = y$, then $\lim_{n \to \infty} (x_n y_n) = xy$. This fundamental result is a cornerstone of mathematical analysis and has broad implications in various mathematical disciplines. We have explored the definition of sequence convergence, the boundedness of convergent sequences, and the importance of the triangle inequality in constructing the proof. By understanding the steps involved in this proof, we gain a deeper appreciation for the nature of limits and their role in mathematical reasoning. Moreover, we have highlighted common pitfalls and misconceptions that can arise when working with limits, emphasizing the need for a careful and rigorous approach.

The ability to manipulate limits and prove limit theorems is essential for advanced mathematical study. The techniques and concepts discussed in this article provide a solid foundation for further exploration of real analysis, calculus, and related fields. As we continue to delve into the world of mathematics, the understanding of limits will undoubtedly serve as a powerful tool for problem-solving and theoretical development.