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The problem is asking to prove that for any real number \( x \), the floor function (denoted by \( \lfloor x \rfloor \)) satisfies the property:

\[
\lfloor x \rfloor = -\lceil -x \rceil
\]

### Definitions:
- \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \). 
- \( \lceil x \rceil \) is the smallest integer greater than or equal to \( x \).

### Proof:

1. **Start with the definition of the floor function**:  
   For any real number \( x \), \( \lfloor x \rfloor \) is the largest integer such that:
   \[
   \lfloor x \rfloor \leq x < \lfloor x \rfloor + 1
   \]

2. **Consider the ceiling of \( -x \)**:  
   By definition, \( \lceil -x \rceil \) is the smallest integer such that:
   \[
   \lceil -x \rceil \geq -x > \lceil -x \rceil - 1
   \]

3. **Negate the inequality**:  
   Multiply the inequality by \(-1\) to reverse the inequality signs:
   \[
   -\lceil -x \rceil \leq x < -\lceil -x \rceil + 1
   \]

4. **Comparison with the floor function**:  
   The expression \( -\lceil -x \rceil \) fits the definition of \( \lfloor x \rfloor \) as it is the greatest integer less than or equal to \( x \).

Thus, we have shown that:
\[
\lfloor x \rfloor = -\lceil -x \rceil
\]

This completes the proof.

Would you like more details or explanations? Here are some related questions you might find useful:

1. How does the floor function behave with rational versus irrational numbers?
2. Can you derive a similar expression involving the ceiling function \( \lceil x \rceil \)?
3. How does the floor function relate to the fractional part of a number \( x \)?
4. What is the relationship between floor and ceiling functions for positive and negative numbers?
5. How can the floor function be used in defining piecewise functions?

**Tip**: Understanding the relationship between floor and ceiling functions helps in solving many problems related to rounding and integer approximations.

Given a real number x, ⌊x⌋ is the biggest integer that is less than or equal to x; i.e., ⌊x⌋ ≤ x < ⌊x⌋ + 1. Similarly, we have that ⌈x⌉ is the unique integer satisfying ⌈x⌉ - 1 < x ≤ ⌈x⌉. Show that ⌊x⌋ = -⌈-x⌉ for all real numbers x.

Explore the proof showing that for any real number x, the floor function ⌊x⌋ is equal to the negative of the ceiling function ⌈-x⌉. This problem involves understanding the properties of floor and ceiling functions and their relationships with inequalities.

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