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.