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Chapter 1: Problem 13

Prove that \(n<2^{n}\) for all \(n \in \mathbb{N}\).

Short Answer

Expert verified
By applying the principle of mathematical induction, we can prove that \(n<2^{n}\) for all natural numbers \(n\). We first validate the base case for \(n = 1\), then make an inductive assumption that the statement holds true for \(n = k\). Finally, we prove that if the statement is true for \(n = k\), then it also holds true for \(n = k+1\). Hence, the statement is valid for all \(n \in \mathbb{N}\).

Step by step solution

01

Base Case

The base case involves testing the smallest natural number, which is 1. We have to check if the statement holds true for \(n = 1\). So, we test \(1 < 2^{1}\). Since \(1 < 2\), the statement holds true for the base case.
02

Inductive Step (Assumption)

Next we make an inductive assumption, also known as inductive hypothesis. We assume that the statement holds true for some \(n = k\), where \(k\) is a natural number. This gives us: \(k < 2^{k}\).
03

Inductive Step (Proof)

After forming the assumption, we will have to prove that the property holds true for \(n = k+1\). We start with \(k+1 < 2^{k+1}\), then we can write the right side inequality as \(k+1 < 2^{k} * 2\). Since we assumed \(k < 2^{k}\), we know that \(2^{k} > k\), hence \(2^{k} * 2 = 2^{k} + 2^{k} > k + 2 > k + 1\). This proves that if the statement is true for \(n = k\), then it is true for \(n = k+1\). So, by induction, the statement \(n < 2^{n}\) is true for all \(n \in \mathbb{N}\).

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Key Concepts

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

Real Analysis
Real analysis is a significant area of mathematical study that focuses on the behavior and properties of real numbers, sequences, series, and real-valued functions. Central to real analysis is the understanding of limits, continuity, differentiation, and integration. These concepts help in discussing and solving problems related to convergence, divergence, and the approximation of functions.

In the context of the exercise, real analysis is applied to explore inequalities involving real numbers. The exercise provided goes beyond the realm of algebra and ventures into real analysis by examining the growth rate of functions. In this case, the function in question is the exponential function, defined for natural numbers. Understanding how functions grow, what their limits are, and how they behave is foundational in real analysis.

For instance, in the textbook solution, comparing the growth rates of the linear function represented by the natural number 'n' and the exponential function given as '2^n' demonstrates a fundamental concept in real analysis: exponential functions grow much faster than linear ones as 'n' increases. This is an insight that can have important ramifications in various fields such as computer science, economics, and beyond.
Proof Techniques
Proof techniques are the tools that mathematicians use to demonstrate the truth or falsity of mathematical statements. Common proof techniques include direct proofs, proof by contradiction, proof by contrapositive, and mathematical induction, which is used in the exercise.

Mathematical induction is particularly powerful for proving statements about natural numbers. It consists of two main steps: the base case and the inductive step. As seen in the step-by-step solution, the base case verifies that the statement holds for the initial value of the series. The inductive step, on the other hand, demonstrates that if the statement holds for one case, then it holds for the next case as well.

This approach is analogous to dominoes falling; if the first domino falls (base case) and each domino is guaranteed to knock over the next one (inductive step), then all the dominoes will fall. Mathematical induction is a cornerstone in proving theorems in various branches of mathematics, particularly those dealing with the properties of numbers and sequences.
Natural Numbers
Natural numbers, denoted as \( \mathbb{N} \), refer to the set of positive integers starting from 1 and moving forward indefinitely (1, 2, 3, ...). They are the most basic and familiar numbers in mathematics and are used to count and order objects in the real world.

In the exercise, we are dealing with properties of natural numbers and their relationship to exponential expressions. It's interesting to note that natural numbers can be aspects of simple counting problems, but they also appear in more complex mathematical discussions, such as the one presented in the exercise where the behavior of an inequality involving natural numbers and exponents is examined.

The concept of natural numbers is foundational for all higher mathematics. From counting to constructing proofs involving sequences and series, as in mathematical induction, natural numbers are indispensable tools. In real analysis, the properties of these numbers help to define more complex structures and theories, showcasing their critical role in mathematics.

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Most popular questions from this chapter

(a) Show that if \(f: A \rightarrow B\) is injective and \(E \subseteq A\), then \(f^{-1}(f(E))=E\). Give an example to show that equality need not hold if \(f\) is not injective. (b) Show that if \(f: A \rightarrow B\) is surjective and \(H \subseteq B\), then \(f\left(f^{-1}(H)\right)=H\). Give an example to show that equality need not hold if \(f\) is not surjective.

Let \(f: A \rightarrow B\) and \(g: B \rightarrow C\) be functions. (a) Show that if \(g \circ f\) is injective, then \(f\) is injective. (b) Show that if \(g \circ f\) is surjective, then \(g\) is surjective.

Conjecture a-formula for the sum \(1 / 1 \cdot 3+1 / 3 \cdot 5+\cdots+1 /(2 n-1)(2 n+1)\), and prove your conjecture by using Mathematical Induction.

(a) Suppose that \(f\) is an injection. Show that \(f^{-1} \circ f(x)=x\) for all \(x \in D(f)\) and that \(f \circ f^{-1}(y)=y\) for all \(y \in R(f)\) (b) If \(f\) is a bijection of \(A\) onto \(B\), show that \(f^{-1}\) is a bijection of \(B\) onto \(A\).

Prove that \(5^{2 n}-1\) is divisible by 8 for all \(n \in \mathbb{N}\)
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