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Chapter 5: Problem 27

Show that if the statement \(P(n)\) is true for infinitely many positive integers \(n\) and \(P(n+1) \rightarrow P(n)\) is true for all positive integers \(n,\) then \(P(n)\) is true for all positive integers \(n .\)

Short Answer

Expert verified
\(P(n)\) is true for all positive integers \(n\).

Step by step solution

01

- Understand the Given Conditions

The problem provides two conditions: (1) The statement \(P(n)\) is true for infinitely many positive integers \(n\). (2) The statement \(P(n+1) \rightarrow P(n)\) is true for all positive integers \(n\). The goal is to show that \(P(n)\) is true for all positive integers \(n\).
02

- Analyze the Implication

Given that \(P(n+1) \rightarrow P(n)\) is true for all positive integers \(n\), we can rewrite this as: if \(P(n+1)\) is true, then \(P(n)\) must also be true. This means if the statement is true for any integer, it is also true for the previous integer.
03

- Consider the Infinitely Many True Values

Since \(P(n)\) is true for infinitely many positive integers, we know there is at least one integer, let's say \(k\), for which \(P(k)\) is true. If we choose the largest such \(k\), \(P(k)\) will be true.
04

- Use the Implicative Chain

From the given condition and the largest integer \(k\) for which \(P(k)\) is true, we can repeatedly apply the implication \(P(n+1) \rightarrow P(n)\). Thus, \(P(k)\) being true implies \(P(k-1)\) must be true, then \(P(k-2)\) must be true and so on, down to any positive integer.
05

- Generalize the Result

Given that \(P(k)\) is true for an infinitely large \(k\), using our implicative chain argument, it follows that \(P(m)\) must be true for any positive integer \(m\) less than or equal to \(k\). Since \(k\) is infinitely large, it covers all positive integers.

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Key Concepts

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

implication
An implication in logic connects two statements in such a way that if the first statement (the hypothesis) is true, then the second statement (the conclusion) must also be true. This is denoted by the symbol \rightarrow. In our exercise, we have the implication \(P(n+1) \rightarrow P(n)\) which means 'if \(P(n+1)\) is true, then \(P(n)\) must also be true.' This relationship helps us understand how the truth of the statement for one value can affect the truth for the previous value. Implications are foundational in mathematical proofs to build a chain of logical conclusions from one statement to another.
infinite set
An infinite set is a set that has an unending number of elements. In our scenario, we consider that the statement \(P(n)\) holds true for 'infinitely many' positive integers. This means there is no upper limit to the values of \(n\) for which \(P(n)\) is true. The concept of an infinite set is important because it allows us to use infinitely large integers to establish general truths about all positive integers. In infinite sets, patterns and behaviors observed in large members of the set often generalize to all members. Therefore, once we identify any infinitely large \(k\) where \(P(k)\) is true, we understand that the statement must also be true for all integers leading up to it.
proof by contradiction
Proof by contradiction is a technique where we assume the opposite of what we want to prove and then show this assumption leads to a contradiction. Let's apply this to our problem: Our goal is to show \(P(n)\) is true for all positive integers. Assume there exists some smallest positive integer \(j\) for which \(P(j)\) is false. Then, given our conditions, \(P(j+1)\) must also be false because if \(P(j+1)\) was true, then it would imply \(P(j)\) is true, contradicting our assumption that \(P(j)\) is false. Continuing this way, we find that \(P(n)\) would be false for any larger integer, contradicting our initial statement that \(P(n)\) is true for infinitely many positive integers. Therefore, our assumption must be wrong, and \(P(n)\) must be true for all positive integers.

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