The task in the image asks to prove that the interval $[0, 1]$ is not countable. Here's a structured proof of that statement using Cantor's diagonalization argument.

Proof (Cantor's Diagonalization Argument):

1. Assume for contradiction: Suppose the interval $[0, 1]$ is countable. This would mean we can list all the real numbers in this interval as a sequence: $x_1, x_2, x_3, \dots$ where each $x_i$ is a real number in $[0, 1]$, and every real number in this interval appears somewhere on this list.

2. Decimal expansion: Write each number in its decimal expansion. That is, for each $x_i$, we represent it as: $x_1 = 0.a_{11}a_{12}a_{13}\dots$ $x_2 = 0.a_{21}a_{22}a_{23}\dots$ $x_3 = 0.a_{31}a_{32}a_{33}\dots$ where $a_{ij}$ represents the $j$-th decimal digit of the $i$-th number.

3. Construct a new number: We now construct a new number $y$ in the interval $[0, 1]$, which is not in the list. Define $y = 0.b_1b_2b_3\dots$ by choosing its decimal digits as follows:

   • Let $b_1$ be different from $a_{11}$.
   • Let $b_2$ be different from $a_{22}$.
   • Let $b_3$ be different from $a_{33}$, and so on.

   More formally, for each $i$, choose $b_i \neq a_{ii}$, ensuring that $y$ differs from each $x_i$ in at least the $i$-th decimal place.

4. Contradiction: The number $y$ cannot be on the list, because it differs from $x_1$ in the first decimal place, from $x_2$ in the second decimal place, and so on. This contradicts the assumption that all real numbers in $[0, 1]$ are listed.

5. Conclusion: Therefore, the interval $[0, 1]$ is uncountable.

Would you like further details or have any questions about this proof? Here are some related questions to deepen understanding:

1. What does it mean for a set to be countable?
2. Can the set of rational numbers in $[0, 1]$ be counted, and how?
3. How does Cantor's diagonalization argument apply to other infinite sets?
4. What is the difference between countably infinite and uncountably infinite sets?
5. Can you apply diagonalization to other intervals, such as $(0, 1)$?

Tip: The essence of Cantor's diagonalization is creating something new by ensuring it differs at least in one position from every item in a list.