To prove that every real number in the set $(0, 1)$ is an interior point of $(0, 1)$, we can rely on the definition of an interior point.

Definition of an Interior Point:

A point $x$ is said to be an interior point of a set $S$ if there exists an open interval around $x$ that is entirely contained within $S$. In other words, there is some $\varepsilon > 0$ such that the interval $(x - \varepsilon, x + \varepsilon)$ is a subset of $S$.

Proof:

1. Let $x \in (0, 1)$:

   • Consider any point $x \in (0, 1)$. By definition, $x$ is a real number such that $0 < x < 1$.

2. Find an $\varepsilon$ such that the interval $(x - \varepsilon, x + \varepsilon) \subset (0, 1)$:

   • Since $x$ is strictly between 0 and 1, we know that there is some positive distance between $x$ and the endpoints of the interval 0 and 1.
   • Specifically, the distance from $x$ to 0 is $x - 0 = x$, and the distance from $x$ to 1 is $1 - x$.
   • Let $\varepsilon = \min(x, 1 - x)$. This ensures that $\varepsilon$ is the smaller of the distances between $x$ and the two endpoints of the interval (0 and 1).

3. Verify that the interval $(x - \varepsilon, x + \varepsilon)$ lies within $(0, 1)$:

   • By construction, $\varepsilon \leq x$ and $\varepsilon \leq 1 - x$.
   • Therefore:
     • $x - \varepsilon \geq 0$ (so the left endpoint of the interval is greater than or equal to 0).
     • $x + \varepsilon \leq 1$ (so the right endpoint of the interval is less than or equal to 1).
   • This shows that the open interval $(x - \varepsilon, x + \varepsilon)$ is completely contained within $(0, 1)$.

4. Conclusion:

   • Since for every $x \in (0, 1)$, we can find an open interval $(x - \varepsilon, x + \varepsilon) \subset (0, 1)$, we conclude that every point in $(0, 1)$ is an interior point of $(0, 1)$.

Thus, every real number in $(0, 1)$ is an interior point of $(0, 1)$.

Would you like further details on any part of this explanation?

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Here are 5 related questions to expand on this topic:

1. How does the definition of an open set relate to interior points?
2. Can you prove whether $0$ or $1$ are interior points of $[0, 1]$?
3. How would you prove that a closed interval, like $[0, 1]$, has no interior points at its boundary?
4. What is the relationship between interior points and boundary points of a set?
5. How does the concept of interior points apply to higher-dimensional spaces, such as in $\mathbb{R}^2$?

Tip: When proving properties about sets and intervals, always start with the definitions and verify conditions step by step.