The question asks to "Prove that the interval $[0,1]$ is not countable."

Here’s a proof using Cantor's Diagonalization Argument, a classic method to show that the interval $[0,1]$ is uncountable:

Proof:

1. Assume for contradiction that the interval $[0,1]$ is countable. This means that the real numbers in $[0,1]$ can be listed in a sequence $x_1, x_2, x_3, \dots$.

2. Each number in the interval $[0,1]$ can be represented by its decimal expansion: $x_1 = 0.a_{11}a_{12}a_{13}\dots$ $x_2 = 0.a_{21}a_{22}a_{23}\dots$ $x_3 = 0.a_{31}a_{32}a_{33}\dots$ $\vdots$ Where $a_{ij}$ represents the $j$-th digit of the $i$-th number.

3. Construct a new number $y = 0.b_1b_2b_3 \dots$ where each digit $b_i$ is chosen such that $b_i$ is different from the $i$-th digit of $x_i$. For example, let:

   1 & \text{if } a_{ii} \neq 1 \\ 2 & \text{if } a_{ii} = 1 \end{cases}$$ This ensures that $$y$$ is not equal to $$x_i$$ for any $$i$$, because $$y$$ differs from each $$x_i$$ at least in the $$i$$-th decimal place.

4. Since $y$ differs from every number in the supposed list, $y$ is a real number in $[0,1]$ that is not in the list $x_1, x_2, x_3, \dots$.

5. This is a contradiction because we assumed that all numbers in $[0,1]$ were listed.

6. Therefore, the interval $[0,1]$ cannot be countable. Hence, $[0,1]$ is uncountable.

This proof shows that it is impossible to form a one-to-one correspondence between the real numbers in $[0,1]$ and the natural numbers, implying that $[0,1]$ is uncountable.

Would you like further clarification or details on this proof?

Related Questions:

1. What is the difference between countable and uncountable sets?
2. How does Cantor's diagonalization work for proving other sets are uncountable?
3. Can you provide examples of countable sets besides the natural numbers?
4. What are some practical implications of uncountable sets in mathematics?
5. How does the cardinality of the interval $[0, 1]$ compare to the set of real numbers?

Tip:

When dealing with infinite sets, Cantor's diagonalization is a powerful method to demonstrate uncountability.