To prove that one of the statements $x < 0$, $x = 0$, or $x > 0$ is always true for any real number $x$, we need to rely on the trichotomy property of real numbers.

Proof:

The trichotomy property states that for any real number $x$, exactly one of the following three conditions holds:

1. $x$ is negative: $x < 0$
2. $x$ is zero: $x = 0$
3. $x$ is positive: $x > 0$

These three cases cover all possible values for a real number $x$. This means that for any given real number, $x$, it must fall into exactly one of these categories.

• If $x = -2$, for example, then $x < 0$.
• If $x = 3$, then $x > 0$.
• If $x = 0$, then $x = 0$.

Thus, one of these statements will always be true for any real number $x$.

Conclusion:

For every real number $x$, one (and only one) of the conditions $x < 0$, $x = 0$, or $x > 0$ is always true, by the trichotomy property of real numbers.

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Would you like a deeper explanation or have any questions on this?

Here are 5 related questions to explore:

1. What is the law of trichotomy and how does it differ from other properties of real numbers?
2. How is the trichotomy property used in proofs of inequalities?
3. Can you prove similar properties for complex numbers?
4. How does the trichotomy property apply in practical real-world problems?
5. Why can't a real number $x$ satisfy more than one of the conditions at the same time?

Tip: The trichotomy property is foundational in real analysis and helps ensure that we can define order relations consistently on real numbers.