Problem 1 - A Primer of Real Analysis

Question

Suppose \(\left\\{a_{i}\right\\}_{i \in I}\) and \(\left\\{b_{i}\right\\}_{i \in J}\) are sequences in \(\mathbb{Q}\) with $$\lim _{i \rightarrow \infty} a_{i}=\lim _{i \rightarrow \infty} b_{i}$$ Show that \(a_{i} \sim b_{i}\).

Step-by-Step Solution

Answer

The sequences \(a_i\) and \(b_i\) are asymptotically equivalent as \( \lim_{i \to \infty} \frac{a_i}{b_i} = 1 \).

Step 1: Understanding the Concept of Asymptotic Equivalence

Two sequences \( \{a_i\} \) and \( \{b_i\} \) are asymptotically equivalent, denoted \( a_i \sim b_i \), if \( \lim_{i \to \infty} \frac{a_i}{b_i} = 1 \). To prove this, we must analyze the behavior of \( \frac{a_i}{b_i} \) as \( i \) approaches infinity.

Step 2: Applying Limit Definition

Given that \( \lim_{i \to \infty} a_i = L \) and \( \lim_{i \to \infty} b_i = L \) in \( \mathbb{Q} \), both sequences converge to the same limit \( L \). We need to show \( \lim_{i \to \infty} \frac{a_i}{b_i} = 1 \).

Step 3: Manipulating Sequence Terms

Since \( a_i \) and \( b_i \) both approach \( L \), for any small \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( i > N \), \(|a_i - L| < \epsilon L \) and \(|b_i - L| < \epsilon L\). Thus, both \( a_i \) and \( b_i \) are close enough to \( L \) when \( i \) is sufficiently large.

Step 4: Exploring the Limit of the Ratio

Considering the ratio \( \frac{a_i}{b_i} \), we have:\[\frac{a_i}{b_i} = \frac{L + (a_i - L)}{L + (b_i - L)}\]As \( i \rightarrow \infty \), \( a_i - L \to 0 \) and \( b_i - L \to 0 \). Therefore, the ratio approximates:\[\frac{L + 0}{L + 0} = 1\]

Step 5: Using the Limit Laws

By the Sandwich Theorem (Squeeze Theorem), since \( \lim_{i \to \infty} a_i = L \) and \( \lim_{i \to \infty} b_i = L \), their ratio approaches \( 1 \) as \( i \to \infty \). Thus, we can conclude that:\[\lim_{i \to \infty} \frac{a_i}{b_i} = 1\]Hence, \( a_i \sim b_i \).

Key Concepts

Limits

Limits are fundamental in calculus and mathematical analysis. They describe how a function or sequence behaves as it approaches a specific point or infinity. In our context, we are interested in the limits of sequences. When we say that the limit of sequence \( \{a_i\} \) as \( i \to \infty \) is \( L \), we mean that as \( i \) becomes very large, the terms \( a_i \) get closer and closer to \( L \).

• The notation \( \lim_{i \to \infty} a_i = L \) denotes this behavior.
• A limit is essential in determining the convergence of sequences, where a sequence converges if it approaches a specific limit.

Understanding limits helps us comprehend the concept of asymptotic equivalence, where two sequences are said to be asymptotically equivalent if the ratio of their terms approaches 1 as \( i \) goes to infinity.

Sequences

A sequence is an ordered list of numbers, each number in the list is called a term. Sequences can be finite or infinite. In mathematical analysis, we often deal with infinite sequences.

• Sequences are usually written as \( \{a_i\} \) where \( i \) is the index running over the natural numbers.
• Each term \( a_i \) is defined by a specific rule or formula.
• The behavior of sequences as \( i \to \infty \) is usually studied to determine their limit or to understand their asymptotic properties.

In this exercise, \( \{a_i\} \) and \( \{b_i\} \) are sequences in \( \mathbb{Q} \) – the set of rational numbers. Both sequences are shown to converge to a common limit \( L \). This common behavior near their limit is crucial to showing that \( a_i \sim b_i \).

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a vital tool in calculus used to find the limit of a function or sequence by comparing it with two other functions or sequences whose limits are known and are equal.

• The theorem states that if \( a_i \le c_i \le b_i \) for all sufficiently large \( i \), and \( \lim_{i \to \infty} a_i = \lim_{i \to \infty} b_i = L \), then \( \lim_{i \to \infty} c_i = L \) as well.
• This theorem is especially useful when directly finding the limit of \( \{c_i\} \) is hard, but \( \{c_i\} \) is bound by two sequences that converge to the same limit.
• In our exercise, the Squeeze Theorem helps confirm that \( \lim_{i \to \infty} \frac{a_i}{b_i} = 1 \), given that both sequences \( \{a_i\} \) and \( \{b_i\} \) converge to the same limit \( L \).

Using this theorem, we ensure that the asymptotic equivalence condition \( a_i \sim b_i \) holds true, tying together the concepts of limits and sequences seamlessly.