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Chapter 2: Problem 14

If \(y>0\), show that there exists \(n \in \mathbb{N}\) such that \(1 / 2^{n}

Short Answer

Expert verified
Let's denote \(n'\) as the smallest integer not smaller than \(1/y\). We define \(n = n'\). By the definition of \(n'\), we have \(1/y \leq n \leq n+1\). This means that we have \(1/y \leq n\), so \(y \geq 1/n\), and in terms of powers of two, we have \(1/2^n \leq y\). Therefore, for any real number \(y > 0\), there exists a natural number \(n\) such that \(1/2^n < y\).

Step by step solution

01

Understanding the Problem

This is an exercise in using the Archimedean property of the real numbers. We are asked to prove that, given any real number \(y > 0\), there exists a natural number \(n\) such that \(1/2^n < y\).
02

Proving the inequality

Let's try to prove the inequality \(1/2^n < y\) from the right. We should find such \(n\) that \(1/y < 2^n\). To do this, we can apply the Archimedean property of real numbers which states that given any positive real number \(x\), there exists a natural number \(n\) greater than \(x\). Hence, since \(1/y>0\), there exists \(n\) such that \(1/y < n\). We'll denote it as \(n'\).
03

Finding the exact value for n

But we need to find \(n\) that is larger than \(1/y\) but also a power of 2. This is because, in terms of our inequality, \(n\) must be such a number that \(2^n\) is larger than \(1/y\). We know that \(2^n\) grows faster than \(n\) itself, thus there exists such \(n\) that \(1/y < 2^n\). Let's find it: we can take \(n\) to be the smallest integer that is not less than \(n'\). This means that \(n\) satisfies both our conditions: \(n > n'\) and \(n\) is a natural number.

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Key Concepts

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

Real Numbers
Real numbers are numbers that include all the rational and irrational numbers. They are continuous and can be found on an infinite line called the real number line. Real numbers are not limited to whole numbers, fractions, or decimals, but they also include irrational numbers like \( \sqrt{2} \) and \( \pi \).
  • A rational number is any number that can be expressed as a fraction \( \frac{a}{b} \) where both \( a \) and \( b \) are integers and \( b eq 0 \).
  • An irrational number cannot be made into a fraction, such as \( \sqrt{2}, \) \( \pi, \) or \( e \).
  • All these numbers together make up the set of real numbers.
In mathematics, the real numbers are crucial because they allow for a consistent way to discuss sizes and quantities, important in both algebra and calculus.
Understanding real numbers is essential for complex mathematical concepts, including proving inequalities.
Inequality Proof
In mathematics, an inequality proofs involves showing that one quantity is larger or smaller than another. In the original exercise, the proof demonstrates that there exists a natural number \( n \) such that \( 1/2^n < y \), where \( y \) is any positive real number.
  • To address an inequality, we often need to manipulate or rearrange it to arrive at a solution that satisfies the condition logically and mathematically.
  • In our case, utilizing the Archimedean property of real numbers simplifies proving the inequality.
The strategy mainly involves finding \( n \) to satisfy both the terms of powers of two and real numbers, ensuring the inequality holds true. Practicing such proofs sharpens logical thinking and problem-solving skills.
Natural Numbers
Natural numbers are the set of numbers used for counting and ordering, beginning from 1 and continuing infinitely: \( 1, 2, 3, 4, \ldots \). They are a subset of real numbers but don't include negative numbers, fractions, or decimals.
  • Natural numbers are typically represented by the symbol \( \mathbb{N} \).
  • These numbers are essential in various mathematical operations and form the basis of simple arithmetic.
In the context of the given problem, we need to find a natural number \( n \) such that \( 1/2^n < y \). This demonstrates the bridge between the set of natural numbers and the infinite continuity of real numbers when exploring properties like the Archimedean property.
Power of Two
The power of two refers to numbers that can be expressed as \( 2^n \) where \( n \) is a natural number. They are integral in computing and digital systems, given their binary nature.
  • Examples include \( 2, 4, 8, 16, 32, \ldots \).
  • Powers of two are a rapidly growing sequence and are crucial for exploring concepts such as exponential growth or decay.
In the inequality proof from the original exercise, the focus is to find a suitable power of two that surpasses a given \( 1/y \), a factor critical in the shown approach. Understanding how powers of two behave can be highly beneficial when engaging in mathematical proofs and real-world applications.

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Most popular questions from this chapter

Let \(S_{1}:=\\{x \in \mathbb{R}: x \geq 0\\} .\) Show in detail that the set \(S_{1}\) has lower bounds, but no upper bounds. Show that inf \(S_{1}=0\).

If \(a, b \in \mathbb{R}\), show that \(|a+b|=|a|+|b|\) if and only if \(a b \geq 0\).

Find the infimum and supremum, if they exist, of each of the following sets. (a) \(\quad A:=\\{x \in \mathbb{R}: 2 x+5>0\\}\) (b) \(\quad B:=\left\\{x \in \mathbb{R}: x+2 \geq x^{2}\right\\}\) (c) \(\quad C:=\\{x \in \mathbb{R}: x<1 / x\\}\) (d) \(D:=\left\\{x \in \mathbb{R}: x^{2}-2 x-5<0\right\\}\).

If \(a \neq 0\) and \(b \neq 0\), show that \(1 /(a b)=(1 / a)(1 / b)\).

Let \(K_{n}:=(n, \infty)\) for \(n \in \mathbb{N}\). Prove that \(\bigcap_{n=1}^{\infty} K_{n}=\emptyset\).
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