To show that the open interval $(0, 1)$ is uncountable, we can use Cantor's diagonalization argument, a classic proof method. Here's the proof:

Proof:

1. Assumption for Contradiction: Assume that the interval $(0, 1)$ is countable. This means that we can list all the numbers in $(0, 1)$ in a sequence $x_1, x_2, x_3, \ldots$. Each of these numbers can be represented by an infinite decimal expansion.

2. Decimal Representation: Let the decimal expansion of $x_i$ be written as: $x_i = 0.a_{i1}a_{i2}a_{i3}\ldots$ where $a_{ij}$ is the $j$-th digit in the decimal expansion of $x_i$.

3. Construct a New Number: We will construct a new number $y$ in $(0, 1)$ that is not in the list $x_1, x_2, x_3, \ldots$ by ensuring that it differs from each $x_i$ in at least one decimal place.

   Define the decimal expansion of $y$ as follows:

   • Let the $j$-th digit of $y$ be $b_j$, where: \begin{cases} 1 & \text{if } a_{jj} \neq 1 \\ 2 & \text{if } a_{jj} = 1 \end{cases}$$

   This construction ensures that the $j$-th digit of $y$ differs from the $j$-th digit of $x_j$. Specifically, if the $j$-th digit of $x_j$ is 1, then the $j$-th digit of $y$ will be 2, and vice versa.

4. Conclusion: The number $y$ constructed in this way differs from each $x_i$ in the $i$-th decimal place. Therefore, $y$ cannot be equal to any $x_i$ in the list. This means $y$ is not in the list $x_1, x_2, x_3, \ldots$.

Since we assumed that $(0, 1)$ was countable, but found a number not in our list, we arrive at a contradiction. Hence, our initial assumption was false.

Conclusion:

The open interval $(0, 1)$ is uncountable.

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Would you like further details or have any questions on this topic?

Related Questions:

1. What is the definition of countable and uncountable sets?
2. How does Cantor's diagonalization argument work in other contexts?
3. Can you give an example of a countable set?
4. What is the difference between countable and finite sets?
5. How do we prove that the set of real numbers is uncountable?
6. What are the implications of a set being uncountable?
7. What is the cardinality of the set of real numbers?
8. How do we define different sizes of infinity?

Tip: The construction of a new number in Cantor's diagonalization argument is key in demonstrating uncountability. It highlights the impossibility of listing all elements of the set.