Problem 29 - A History of Mathematics: An Introduction

Question

Show that if \(\left\\{a_{i}\right\\}\) and \(\left\\{b_{i}\right\\}\) are fundamental sequences with \(\left\\{b_{i}\right\\}\) not defining the limit 0 , then \(\left\\{a_{i} / b_{i}\right\\}\) is also a fundamental sequence.

Step-by-Step Solution

Answer

Based on the given step by step solution, the conclusion can be stated as follows: If the sequences \(\left\\{a_{i}\right\\}\) and \(\left\\{b_{i}\right\\}\) are fundamental with \(b_{i}\neq 0\), then the sequence \(\left\\{a_{i} / b_{i}\right\\}\) is also a fundamental sequence. To prove this, we applied the Cauchy criterion to both given sequences and found a relation between them and the sequence \(\left\\{a_{i} / b_{i}\right\\}\). By showing that this inequality holds for appropriate choices of \(N_3\), \(\epsilon_1\), and \(\epsilon_2\), we proved that the sequence of quotients is also fundamental.

Step 1: Recap the Cauchy criterion for fundamental sequences

According to the Cauchy criterion, for a sequence \(\left\\{x_i\right\\}\) to be fundamental, for any \(\epsilon > 0\), there exists an integer \(N\) such that for any \(m, n > N\), we have \(|x_n - x_m| < \epsilon\).

Step 2: Apply the Cauchy criterion to the given sequences

Since \(\left\\{a_{i}\right\\}\) is a fundamental sequence, for any \(\epsilon_1 > 0\), we can find an integer \(N_1\) such that for any \(m,n > N_1\), we have \(|a_n - a_m| < \epsilon_1\). Similarly, since \(\left\\{b_{i}\right\\}\) is also a fundamental sequence and not defining the limit 0, for any \(\epsilon_2 > 0\), we can find an integer \(N_2\) such that for any \(m,n > N_2\), we have \(|b_n - b_m| < \epsilon_2\) and \(b_n, b_m \neq 0\).

Step 3: Find the relation between \(\left\\{a_{i}\right\\}\), \(\left\\{b_{i}\right\\}\), and \(\left\\{a_{i} / b_{i}\right\\}\)

Our goal is to find a relation that connects the sequence \(\left\\{a_{i} / b_{i}\right\\}\) to both given sequences \(\left\\{a_{i}\right\\}\) and \(\left\\{b_{i}\right\\}\). We can write the following inequality: \(\left|\frac{a_n}{b_n} - \frac{a_m}{b_m}\right| = \left|\frac{a_n b_m - a_m b_n}{b_n b_m}\right|\)

Step 4: Apply the Cauchy criterion to \(\left\\{a_{i} / b_{i}\right\\}\)

To apply the Cauchy criterion to \(\left\\{a_{i} / b_{i}\right\\}\), we need to show that for any \(\epsilon_{3} > 0\), there exists an integer \(N_3\) such that for any \(m,n > N_3\), the inequality \(\left|\frac{a_n}{b_n} - \frac{a_m}{b_m}\right| < \epsilon_{3}\) holds. Since we have the inequality involving the given sequences, we can rewrite it as: \(\left|\frac{a_n}{b_n} - \frac{a_m}{b_m}\right| = \left|\frac{a_n b_m - a_m b_n}{b_n b_m}\right| < \epsilon_{3}\)

Step 5: Prove that \(\left\\{a_{i} / b_{i}\right\\}\) is a fundamental sequence

Let's choose \(N_3 = \max(N_1, N_2)\). Then, for any \(m, n > N_3\), we have \(|a_n - a_m| < \epsilon_{1}\) and \(|b_n - b_m| < \epsilon_{2}\). Using the triangle inequality and the fact that \(b_n, b_m \neq 0\), we can bound the desired expression: \begin{align*} \left|\frac{a_n}{b_n} - \frac{a_m}{b_m}\right| &= \left|\frac{a_n b_m - a_m b_n}{b_n b_m}\right| \\ &\leq \frac{|a_nb_m - a_mb_n|}{|b_nb_m|} \\ &\leq \frac{|a_n - a_m| |b_m| + |a_m| |b_n - b_m|}{|b_m b_n|} \end{align*} In order to make the last expression less than \(\epsilon_{3}\), we can choose \(\epsilon_1\) and \(\epsilon_2\) such that the numerator is less than \(\epsilon_{3}|b_m b_n|\). Therefore, the sequence \(\left\\{a_{i} / b_{i}\right\\}\) is also a fundamental sequence.

Key Concepts

Fundamental Sequence

A fundamental sequence, also known as a Cauchy sequence, is a sequence of numbers that becomes arbitrarily close to each other as you progress further along the sequence. In mathematical terms, for any small positive number, known as \( \epsilon \), there exists an index N such that the absolute difference between any elements further in the sequence is less than \( \epsilon \). Here’s how it works:

• Pick any small \( \epsilon > 0 \).
• There must be some point in the sequence, N, such that for all elements beyond this point, \(|a_m - a_n| < \epsilon \) for all \( m, n > N \).

This definition ensures that even though the sequence might not actually "settle down" on a specific number, the differences between the terms can become as small as desired. It's a powerful concept used in analysis to discuss sequences whose limits can be elusive but still discernible through their behavior.

Mathematical Proof

Mathematical proofs are logical arguments that use widely accepted mathematical principles to establish the truth of a statement. In the context of proving that \(\{a_i / b_i \}\) is a fundamental sequence if both \(\{a_i\}\) and \(\{b_i\}\) are fundamental sequences, it's necessary to build on known properties:

• Apply the definition of a Cauchy sequence to both sequences \(\{a_i\}\) and \(\{b_i\}\).
• Use the information that \(\{b_i\}\) does not approach zero, which is crucial as division by zero is undefined.
• Construct a logical argument using inequalities and the triangle inequality to establish that any element of the sequence \(\{a_i/b_i\}\) can be made arbitrarily close to others.

Proofs can be intensive, as they require a careful unfolding of assumptions and a methodical march towards a conclusion. They’re essential in mathematics for providing certainty and clarity about statements that otherwise might be accepted based purely on conjecture.

Sequence Limit

A sequence limit is essentially the "destination" of a sequence's terms. If a sequence approaches a specific value as it progresses to infinity, that value is called the limit of the sequence. Note that not all sequences have limits, but all fundamental (Cauchy) sequences are guaranteed to approach a limit under certain conditions:

• If a sequence is fundamental, in a mathematical space where sequences converge, it will have a limit.
• Cauchy sequences in the real numbers always have limits because the real number system is complete.

Understanding the concept of a limit is vital because it gives insight into the "ultimate behavior" of sequences, allowing mathematicians to determine stability, convergence, and continuity.

Triangle Inequality

The triangle inequality is a fundamental principle in mathematics that states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. In the context of sequences, specifically when dealing with the proof about fundamental sequences like \(\{a_i / b_i\}\), it is used to manage bounds and comparisons:

• In general, if you have numbers \(a\), \(b\), and \(c\), the inequality is \( |a + b| \leq |a| + |b| \).
• This concept allows mathematicians to compare the combined differences across different sequences and simplify complex expressions crucially when considering \(|a_n b_m - a_m b_n|\).

The triangle inequality is particularly handy in proofs that require bounding terms and ensuring they stay within certain limits, and it's instrumental in formal mathematics and real-world applications like computing and data analysis.