Proof by Contradiction: Showing a ≤ b when a ≤ b1 for every b1 > b

• Mr Davis 97
In summary: The statement is proven by contradiction using the fact that if a number is less than or equal to all numbers greater than another number, it must also be less than or equal to that number.In summary, the conversation discusses how to show that if a number is less than or equal to all numbers greater than another number, it must also be less than or equal to that number. The solution uses proof by contradiction and the fact that a number cannot simultaneously be greater than and less than another number.
Mr Davis 97

Homework Statement

Let ##a,b \in \mathbb{R}##. Show if ##a \le b_1## for every ##b_1 > b##, then ##a \le b##.

The Attempt at a Solution

We will proceed by contradiction. Suppose that ##a \le b_1## for every ##b_1 > b##, and ##a > b##. Let ##b_1 = \frac{a+b}{2}##. We see that ##b_1>b##: ##a>b \implies \frac{a+b}{2}> b \implies b_1 >b##. Hence, by the hypothesis, it must be true that ##a \le b_1##. But since ##a \le b_1 \implies a \le \frac{a+b}{2} \implies a \le b##, we have reached a contradiction. We simultaneously have that ##a \le b## and ##a >b##. Hence, it must be the case that the original statement is true, and we are done.

Mr Davis 97 said:

Homework Statement

Let ##a,b \in \mathbb{R}##. Show if ##a \le b_1## for every ##b_1 > b##, then ##a \le b##.

The Attempt at a Solution

We will proceed by contradiction. Suppose that ##a \le b_1## for every ##b_1 > b##, and ##a > b##. Let ##b_1 = \frac{a+b}{2}##. We see that ##b_1>b##: ##a>b \implies \frac{a+b}{2}> b \implies b_1 >b##. Hence, by the hypothesis, it must be true that ##a \le b_1##. But since ##a \le b_1 \implies a \le \frac{a+b}{2} \implies a \le b##, we have reached a contradiction. We simultaneously have that ##a \le b## and ##a >b##. Hence, it must be the case that the original statement is true, and we are done.
Is o.k.

Mr Davis 97

1. What is "Proof by Contradiction"?

Proof by contradiction is a method of mathematical proof that involves assuming the opposite of what you are trying to prove and showing that it leads to a contradiction or impossibility. This contradiction then proves that the original statement must be true.

2. How does "Proof by Contradiction" differ from other proof methods?

Unlike other proof methods, such as direct proof or induction, proof by contradiction does not directly prove a statement to be true. Instead, it proves that the statement cannot be false, which in turn proves that it must be true.

3. When is "Proof by Contradiction" useful?

Proof by contradiction is useful when a direct proof or other method is not possible or is too difficult to construct. It is often used for proving negative statements, such as "there is no largest prime number" or "there are infinitely many irrational numbers".

4. What are the steps involved in a "Proof by Contradiction"?

The first step in a proof by contradiction is to assume the opposite of the statement you are trying to prove. Then, using logical reasoning and previously proven facts, you arrive at a contradiction. This contradiction proves that the original statement must be true.

5. Are there any limitations to "Proof by Contradiction"?

While proof by contradiction can be a powerful tool, it is not always applicable. It can only be used to prove statements that are either true or false, and it cannot be used to prove statements that are neither. Additionally, it may not be the most efficient or elegant method for certain proofs.

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