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Magazine Chapter 11: Problem 89
Prove that if \(\lim _{n \rightarrow \infty} a_{n}=0\) and \(\left\\{b_{n}\right\\}\) is bounded, then \(\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=0\)
Short Answer
Expert verified
The product sequence \( \{a_n b_n\} \) converges to 0 as \( n \to \infty \).
Step by step solution
01
Understand the Given Sequence
We start with two sequences: \( \{a_n\} \) and \( \{b_n\} \). It's given that \( \lim_{n \to \infty} a_n = 0 \), which means that as \( n \) becomes very large, \( a_n \) approaches 0.
02
Examination of the Bounded Sequence
The sequence \( \{b_n\} \) is bounded, which means there exists a real number \( M \) such that for all \( n \), \(|b_n| \le M\). This implies that the values of \( b_n \) are confined within a finite range.
03
Apply Limit Properties to the Product Sequence
We are tasked to prove that \( \lim_{n \to \infty} (a_n b_n) = 0 \). According to the properties of limits, if \( a_n \to 0 \) and \( b_n \) is bounded, then their product \( a_n b_n \) should also tend to zero.
04
Use the Formal Definition of Limits
By the definition of limits, given \( \epsilon > 0 \), there exists an \( N \) such that for all \( n > N \), \(|a_n| < \frac{\epsilon}{M}\). Since \(|b_n|\le M\), we have:\[|a_n b_n| = |a_n||b_n| < \frac{\epsilon}{M} \cdot M = \epsilon.\]Thus, for any \( \epsilon > 0 \), there is an \( N \) such that whenever \( n > N \), \(|a_n b_n| < \epsilon\).
05
Conclude the Proof
Since we showed that for any \( \epsilon > 0 \), we can find an \( N \) such that for all \( n > N \), the inequality \(|a_n b_n| < \epsilon\) holds, it follows by the definition of a limit that \( \lim_{n \to \infty} (a_n b_n) = 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit of a sequence
The concept of a limit of a sequence is foundational in calculus and analysis. In essence, when we say a sequence \(\{a_n\}\) has a limit of zero as \(n\) approaches infinity, written as \(\lim_{n \to \infty} a_n = 0\), this means the terms \(a_n\) get arbitrarily close to 0 as \(n\) becomes very large. To put it simply, no matter how small a positive number we choose, there will eventually be a point in the sequence beyond which all terms fall within this small distance from zero.
- This idea helps in detecting convergence behavior of the sequence.
- It provides a mathematical foundation for analyzing how sequences behave in the long run.
Bounded sequences
A bounded sequence is one that remains within a fixed range of values, regardless of how many terms are in the sequence. For the sequence \(\{b_n\}\) to be bounded, there must exist a number \(M\) such that the absolute value of every term in the sequence is less than or equal to \(M\). This means:\[ |b_n| \leq M \quad \text{for all}\ n. \]
- Being bounded does not necessarily imply that the sequence will converge, but rather it will not grow indefinitely large or small.
- Bounded sequences are crucial when discussing the behavior of products of sequences.
Product of sequences
The product of two sequences \(\{a_n\}\) and \(\{b_n\}\) results in a new sequence given by \(\{a_n b_n\}\). The behavior of this resultant sequence can often be deduced by considering the properties of the original sequences. In our exercise, since \(\{a_n\}\) approaches zero and \(\{b_n\}\) is bounded, the product sequence \(\{a_n b_n\}\) must necessarily approach zero. This conclusion is derived from limit properties:
- If \(\lim_{n\to\infty} a_n = 0\) and \(\{b_n\}\) is bounded, then \(\lim_{n\to\infty} (a_n b_n) = 0\).
- The bounded nature of \(\{b_n\}\) ensures that its values do not 'amplify' any value \(a_n\) might have.
Convergence of sequences
Convergence refers to a sequence's tendency to approach a specific value as the number of terms goes to infinity. In the context of the exercise, the sequence \(\{a_n b_n\}\) converges to 0, under the condition that one sequence converges to zero and the other is bounded.
- Convergence requires that for any \(\epsilon > 0\), no matter how small, there exists an \(N\) such that for all \(n > N\), the difference between the sequence terms and the limit is less than \(\epsilon\).
- This formal definition of convergence is what underlies the proof for \(\lim_{n \to \infty} (a_n b_n) = 0\).
