Final answer:
Cantor's diagonalization argument is used to construct a real number not included in a given list by modifying one digit from each number at a specific decimal place to guarantee a unique number. For example, by changing selected digits to 5 or 4, we can create a new number such as 0.559559..., ensuring it is not on the original list.
Step-by-step explanation:
The question involves the construction of real numbers and Cantor's diagonalization argument, which is used to show that the set of all real numbers is uncountable. It requires generating a number not included in a given list by changing one digit from each number of the list at a specific decimal place, thereby creating a new number distinct from any others on the list. For the provided sequence of numbers, these alterations take place at the nth digit of the nth number in the list, ensuring that the created number will differ from each of the numbers in at least one decimal place.
To illustrate, let's use the list provided (a, b, c, d, e) and modify the nth digit of the nth item on the list. If the digit is not 5, we'll change it to 5; if it is 5, we'll change it to 4, thereby ensuring we have a different number. For the lists (a-e), we would change the first digit of the first number (a) from 1 to 5, the second digit of the second number (b) from 4 to 5, the third digit of the third number (c) from 9 to 5, and so on. This process gives us a new number, which for the first 10 digits after the decimal point would be 0.559559.... Continuing this indefinitely, the rest of the number would be determined by applying the same rule to each subsequent number in the original list.
By Cantor's diagonalization, we can be certain that our newly created number is not on the list because it differs from every number on the list at least in one decimal place.