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Chapter 2: Problem 5

Let \(\left\\{x_{n}\right\\}\) be a bounded sequence and let \(\left\\{y_{n}\right\\}\) be a sequence that converges to 0 . Prove that the sequence \(\left\\{x_{n} y_{n}\right\\}\) converges to 0 .

Short Answer

Expert verified
The sequence \(\{x_n y_n\}\) converges to 0 as \(\{y_n\}\) converges to 0 and \(\{x_n\}\) is bounded.

Step by step solution

01

Understand the Given Sequences

We have two sequences, \(\{x_n\}\) and \(\{y_n\}\). The sequence \(\{x_n\}\) is bounded, meaning there exists some constant \(M > 0\) such that \(|x_n| \leq M\) for all \(n\). The sequence \(\{y_n\}\) converges to 0, which means for any \(\epsilon > 0\), there exists a natural number \(N\) such that for all \(n \geq N\), \(|y_n| < \epsilon\).
02

Formulate the Goal

We need to show that the sequence \(\{x_n y_n\}\) converges to 0. This is done by showing that for any \(\epsilon > 0\), there exists a \(K\) such that if \(n \geq K\), then \(|x_n y_n| < \epsilon\).
03

Use the Definition of Converging Sequence

Since \(\{y_n\}\) converges to 0, given any \(\epsilon > 0\), we can choose \(\epsilon' = \frac{\epsilon}{M}\). There exists a natural number \(N\) such that for all \(n \geq N\), \(|y_n| < \epsilon' = \frac{\epsilon}{M}\).
04

Apply Boundedness of \(\{x_n\}\)

Since \(\{x_n\}\) is bounded, \(|x_n| \leq M\) for all \(n\). This allows us to estimate \(|x_n y_n|\):\\(|x_n y_n| = |x_n||y_n| \leq M|y_n|\).
05

Combine Inequalities for Proof

From step 3, for any \(n \geq N\), \(|y_n| < \frac{\epsilon}{M}\). Using the inequality from step 4, we have:\[|x_n y_n| \leq M|y_n| < M\left(\frac{\epsilon}{M}\right) = \epsilon\]Thus, for any \(n \geq N\), \(|x_n y_n| < \epsilon\), which means \(\{x_n y_n\}\) converges to 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bounded Sequences
A bounded sequence is a core concept in mathematical analysis. It refers to a sequence of numbers that stays within a fixed boundary. To be more precise, a sequence \(\{x_n\}\) is considered bounded if there is some constant \(M > 0\) such that, for every term in the sequence, the absolute value \(|x_n|\) is less than or equal to \(M\). This means that, no matter how far you go in the sequence, the values won't suddenly become infinitely large or small.
  • This concept ensures stability within the sequence.
  • It prevents the sequence from "escaping" to infinity.
  • Boundedness is an important property when dealing with limits and convergence.
Essentially, when a sequence is bounded, it means there is a "ceiling" or "floor" that the sequence doesn't surpass. This helps in analyzing sequences to understand their behavior as we look towards their limits.
Convergent Sequences
Convergent sequences are sequences that approach a specific value as they progress to infinity. For example, suppose you have a sequence \(\{y_n\}\) that converges to 0. This means that, as you take more and more terms (as \(n\) becomes very large), those terms get closer and closer to 0.
  • Convergence means "getting closer over time."
  • For \(\{y_n\}\) to converge to 0, we need \(|y_n| < \epsilon\) for any desired small number \(\epsilon > 0\), past a certain stage of the sequence.
    • This is possible because, given \(\epsilon > 0\), there exists a number \(N\) such that for all terms beyond this point \(n \geq N\), the difference from 0 is smaller than \(\epsilon\).
Convergent sequences are fundamental in calculus and analysis, as they indicate stability and predictability as the sequence expands. Understanding how a sequence converges helps in defining the limits in more complex equations.
Epsilon-Delta Definition
The epsilon-delta definition is a mathematical way of formally defining limits. It's crucial in understanding how sequences and functions behave as they approach a certain value. More specifically, it's used to rigorously prove that a sequence converges to a limit.For a sequence \(\{y_n\}\) converging to a value \(L\), the definition states:
  • For any \(\epsilon > 0\) (no matter how small), there exists a positive integer \(N\).
  • For all terms of the sequence where \(n \geq N\), the terms are within \(\epsilon\) units of \(L\).
    • Mathematically, this is written as: \(|y_n - L| < \epsilon\).
This definition is a backbone of calculus because it provides a precise way of talking about limits and continuity. It's powerful because it offers a way to ensure for all intents and purposes, the terms of a sequence or values of a function stay as close as desired to some limit \(L\), past a certain point. This concept is applied in problems involving convergent and bounded sequences, particularly when proving convergence like with the sequence \(\{x_n y_n\}\) converging to 0.

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Most popular questions from this chapter

Prove that each of the following sequences is convergent and find its limit. (a) \(\sqrt{2} ; \sqrt{2 \sqrt{2}} ; \sqrt{2 \sqrt{2 \sqrt{2}}} ; \cdots\) (b) \(1 / 2 ; \frac{1}{2+1 / 2} ; \frac{1}{2+\frac{1}{2+1 / 2}} ; \cdots\)

Let \(a_{1}=\sqrt{2} .\) Define $$ a_{n+1}=\sqrt{a_{n}+2} \text { for } n \geq 1 $$ (a) Prove that \(a_{n}<2\) for all \(n \in \mathbb{N}\). (b) Prove that \(\left\\{a_{n}\right\\}\) is an increasing sequence. (c) Prove that \(\lim _{n \rightarrow \infty} a_{n}=2\).

Prove or disprove the following statements: (a) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are convergent sequences, then \(\left\\{a_{n}+b_{n}\right\\}\) is a convergent sequence. (b) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are divergent sequences, then \(\left\\{a_{n}+b_{n}\right\\}\) is divergent sequence. (c) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are convergent sequences, then \(\left\\{a_{n} b_{n}\right\\}\) is a convergent sequence. (d) If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are divergent sequences, then \(\left\\{a_{n} b_{n}\right\\}\) is a divergent sequence. (e) If \(\left\\{a_{n}\right\\}\) and \(\left\\{a_{n}+b_{n}\right\\}\) are convergent sequences, then \(\left\\{b_{n}\right\\}\) is a convergent sequence. (f) If \(\left\\{a_{n}\right\\}\) and \(\left\\{a_{n}+b_{n}\right\\}\) are divergent sequences, then \(\left\\{b_{n}\right\\}\) is a divergent sequence.

Prove that the sequence \(\left\\{a_{n}\right\\}\) with \(a_{n}=\sin (n \pi / 2)\) is divergent.

Prove that a subset \(A\) of \(\mathbb{R}\) is open if and only if for any \(x \in A,\) there exists \(n \in \mathbb{N}\) such that \((x-1 / n, x+1 / n) \subset A\)
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