Problem 51 - Essential Calculus Early Transcendentals

Question

Prove that if \(\lim _{n \rightarrow \infty} a_{n}=0\) and \(\left\\{b_{n}\right\\}\) is bounded, then \(\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=0\)

Step-by-Step Solution

Answer

Since \(\lim_{n \rightarrow \infty} a_n = 0\) and \(\{b_n\}\) is bounded, \(\lim_{n \rightarrow \infty} (a_n b_n) = 0\).

Step 1: Understand the Given Condition

We have two sequences: \(\{a_{n}\}\) and \(\{b_{n}\}\). The limit of \(\{a_{n}\}\) as \(n\) approaches infinity is 0, i.e., \(\lim_{n \rightarrow \infty} a_{n} = 0\). Additionally, the sequence \(\{b_{n}\}\) is bounded. That means there exists a constant \(M > 0\) such that for all \(n\), \(|b_{n}| \leq M\).

Step 2: Set Up the Objective

Our goal is to show that the limit of the product sequence \(\{a_{n}b_{n}\}\) as \(n\) approaches infinity is 0. In other words, we need to prove that \(\lim_{n \rightarrow \infty} (a_{n} b_{n}) = 0\).

Step 3: Apply the Epsilon-Delta Definition of Limit

To prove that \(\lim_{n \rightarrow \infty} (a_{n} b_{n}) = 0\), we need to show that for any given \(\epsilon > 0\), there exists a positive integer \(N\) such that for all \(n > N\), \(|a_{n}b_{n}| < \epsilon\).

Step 4: Use the Bound of \(b_n\)

Since \(|b_{n}| \leq M\), we have \(|a_{n}b_{n}| = |a_{n}| |b_{n}| \leq M|a_{n}|\) for all \(n\). Our aim now is to manipulate \(|a_{n}|\) to satisfy the epsilon condition.

Step 5: Choose Appropriate \(N\)

Given that \(\lim_{n \rightarrow \infty} a_{n} = 0\), for any \(\epsilon_1 > 0\), there exists a positive integer \(N\) such that for all \(n > N\), \(|a_{n}| < \frac{\epsilon}{M}\). This choice ensures \(|a_{n}b_{n}| \leq M|a_{n}| < M \cdot \frac{\epsilon}{M} = \epsilon\).

Step 6: Conclude the Proof

By our choice of \(N\), for any \(\epsilon > 0\) and all \(n > N\), we have \(|a_{n}b_{n}| < \epsilon\), proving that \(\lim_{n \rightarrow \infty} (a_{n} b_{n}) = 0\).

Key Concepts

Bounded Sequences

A sequence is considered bounded if there exists a real number that serves as an upper limit for the absolute value of every term in the sequence. Let's denote a sequence by \(\{ b_{n} \}\). If this sequence is bounded, then there is a constant \(M > 0\) such that for every \( n \), the inequality \(|b_{n}| \leq M\) holds true.

Bounded sequences are crucial because they help in controlling other sequences, like in our exercise. Here, \( \{ b_{n} \} \) is bounded, indicating its values won't diverge to infinity as \( n \) increases.

Some typical features of bounded sequences:

• They do not have values that grow indefinitely.
• All the terms are contained within a fixed range.
• Knowing a sequence is bounded is essential for various proofs where stability of values is important.

Sequence Convergence

Sequence convergence refers to whether a sequence approaches a specific value as the number of terms becomes very large. A sequence \( \{ a_{n} \} \) converges to a limit \( L \) if for every \( \epsilon > 0 \) (no matter how tiny), there exists a positive integer \( N \) such that for all \( n > N \), the condition \(|a_{n} - L| < \epsilon\) is satisfied. This essentially means that beyond a certain point, the difference between the sequence terms and \( L \) becomes negligibly small.

For instance, in our exercise, \( \{ a_{n} \} \) converges to 0 as \( n \) approaches infinity. Therefore, no matter how small \( \epsilon \) is, we can find a point after which all terms of the sequence are within \( \epsilon \) of 0.

Key aspects of sequence convergence are:

• A convergent sequence will not "bounce around" at infinity; it will settle into a value.
• This property is crucial for proving limits of more complex expressions, like products of sequences.
• Convergence is a foundation in calculus and analysis, crucial for understanding limits, continuity, and derivatives.

Product of Sequences

The product of sequences involves taking two sequences and forming a new sequence from their multiplied terms. Given sequences \( \{ a_{n} \} \) and \( \{ b_{n} \} \), their product sequence \( \{ a_{n} b_{n} \} \) is defined by each term \( a_{n} b_{n} \).

When dealing with the product of sequences, it's often insightful to apply existing properties of each sequence. In our case, we know \( \lim_{n \to \infty}a_{n} = 0 \) and \( \{ b_{n} \} \) is bounded.

The objective in the exercise was to show that \( \lim_{n \to \infty} (a_{n} b_{n}) = 0 \). Using properties of both convergence (of \( a_{n} \)) and boundedness (of \( b_{n} \)), we demonstrated the product also converges to 0.

Important points about the product of sequences are:

• If one sequence converges to 0 and the other is bounded, their product will converge to 0.
• This principle comes from the property that multiplying a small enough number (close to 0) by a bounded number results in a small product.
• Understanding this helps analyze more complex functions in calculus and real analysis.