https://file.aiquickdraw.com/mathbot/file/1727063487128mhl8mbku.jpg

Let's analyze the problem step by step.

The statement to be proved is:
\[
\forall x \in (0, 1), \exists y \in (0, 1) \text{ such that } x < y.
\]

The proof provided follows these steps:

1. **Let \( x \in (0, 1) \) be given**.
   
   The proof chooses \( y = \frac{x + 1}{2} \).

2. **Check that \( y \in (0, 1) \)**:
   
   - First, verify that \( \frac{1}{2} < y \leq 1 \). To see this:
     - Since \( 0 < x < 1 \), we have \( \frac{1}{2} < \frac{x + 1}{2} \leq 1 \).
     - Hence, \( y \in (0, 1) \).

3. **Verify that \( x < y \)**:
   - \( y = \frac{x + 1}{2} \)
   - Clearly, \( \frac{x}{2} < \frac{x + 1}{2} \), so \( x < y \).

The dropdown selections should be as follows based on the analysis:

1. \( \frac{x + 1}{2} \) is correctly chosen.
2. \( \frac{1}{2} \leq y < 1 \), so choose \( \frac{1}{2} \leq y \leq 1 \).
3. \( \frac{x}{2} + \frac{1}{2} \), so the correct value is \( x < y \).

This completes the proof as intended.

Would you like me to go into more details or have any specific questions?

### Here are 5 related questions to expand this topic:

1. What is the significance of choosing \( y = \frac{x + 1}{2} \) in the proof?
2. Can you choose a different expression for \( y \) such that \( x < y \)?
3. How does this proof technique change if the interval was \( (0, 2) \)?
4. How can you generalize this proof for \( x \in (a, b) \) for arbitrary \( a \) and \( b \)?
5. How does this problem connect with concepts of limits in calculus?

### Tip
In proofs involving inequalities, always check boundary cases carefully to ensure your selected values meet all given conditions.

Prove the statement: ∀x ∈ (0, 1), ∃y ∈ (0, 1) such that x < y.

This proof demonstrates that for any x in (0, 1), there exists a y in (0, 1) such that x < y. The proof involves constructing y as (x + 1) / 2 and verifying that x < y, using concepts from inequalities and real analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation