Question

What exactly is wrong with the following "proof" that for every real number \(x \geq 0\). \(x=2 x:\) Suppose the result is true for all real numbers \(y\) where \(0 \leq y0 .\) Then, \(0

Short answer

The proof mistakenly uses induction incorrectly and falsely concludes \( x = 2x \) for all non-negative \( x \).

Step-by-step solution

Understand the mathematical claim

The claim stated in the proof is that for any non-negative real number \( x \), \( x = 2x \). This suggests that every non-negative real number is equal to its double, which is fundamentally false as it implies every number is zero.

Identify assumptions for induction

The proof attempts to use a form of mathematical induction, where the base case is \( x = 0 \) and the inductive step assumes the property is true for numbers less than \( x \). The inductive claim is incorrect since it lacks a valid base case to start from and misuses the inductive step assumption.

Analyze case 1: x = 0

For \( x = 0 \), the calculation \( 2x = 2 \times 0 = 0 = x \) is technically correct. However, it only shows \( x = 2x \) for \( x = 0 \), not for any \( x > 0 \), and doesn't imply the general result for other non-negative numbers.

Analyze case 2: x > 0

The proof incorrectly assumes "by hypothesis" that \( x/2 = x \), which suggests that the property holds for smaller numbers. However, there is no valid proof for \( x/2 = 2(x/2) = x \); it's based on the false assumption about division and multiplication complying in this form.

Examine conclusion misuse

The proof concludes by asserting the strong form of mathematical induction. However, the initial assumption is logically flawed and the inductive step misuses the hypothesis, leading to an incorrect generalization.

Key concepts

Inductive Hypothesis

In mathematical induction, the inductive hypothesis is a crucial step that assumes the statement to be true for a particular case, often denoted as the case when the variable equals some number less than our number of interest. This assumption forms the cornerstone for proving the base case and then accompanying it with the inductive step. For instance, if we are proving a statement that involves some property of a number, the inductive hypothesis allows us to conjecture that if the statement is true for one number, it should also hold for the next number.
In the given exercise, the proof flawed the use of the inductive hypothesis by incorrectly assuming for a real number x that it holds for smaller numbers (x/2). This is a misuse as the inductive hypothesis should not be assumed true for x/2 without a valid initial base case and should align logically sound steps leading to the conclusion.

Base Case

The base case in mathematical induction is the starting point of the proof. It is the simplest instance, often a specific number like zero or one, where the problem statement can be directly verified as true without assuming any other cases. Checking the base case establishes the foundation from which all subsequent steps can logically follow.
In our exercise, they verified that for x = 0, the statement x = 2x holds true because 2 * 0 equals 0. While mathematically correct for this single point, proving for x = 0 doesn’t help verify the logic for any other potential number in the set that might include numbers greater than zero. This base case, therefore, does not suffice to prove the overall erroneous claim x = 2x for all non-negative numbers.

Inductive Step

The inductive step in a proof by induction is crucial because it shows how to use the assumption from the inductive hypothesis to prove the statement for the next number in the sequence. Essentially, it involves showing that if the proposition holds true for a certain number, say k, then it must also hold true for k + 1. This step ensures that the statement is true for all numbers in a progressively larger set.
In our context, the proof attempted an inductive step by purporting that for a positive x, the claim holds true using a smaller x/2. However, this step is misused because the initial claim that x/2 equals x is based on faulty logic derived from the incorrect assumption about division and multiplication properties. Without forming valid logical justification, the induction step collapses, leading to the fallacy that x equals double itself across all real non-negative numbers.