Constructing a Straggler from an Enumeration of R, {x ∊ R | 0 < x < 1}

Constructing a Straggler from an Enumeration of R, {x ∊ R | 0 < x < 1} Using Cantor's Diagonal Method

Theorem 2: Given an enumeration of the set R, {x € R | 0 < x < 1} there exists a real number X, such that X is a straggler.

Proof: Theorem 2 is proved by contradiction. The goal is to prove that there is an X, {X ∊ R | 0 < X < 1} that is not part of the enumeration. The proof begins by assuming the opposite, namely, that all elements of R, {x ∊ R | 0 < x < 1} can be enumerated. If that assumption is true it must be possible to list the real numbers between 0 and 1 in an infinite table with none missing. An example is depicted graphically below:
 

Real Number

0.2734447…

0.3335894…

0.4664479…

0.5552887…

…

 Construct a two column table. Column A contains an enumeration of real numbers between 0 and 1. Column B contains a representative value of X. To create X ensure that for the first row in the table the first digit to the right of the decimal in X differs from the first digit to the right of the decimal in the real number column. Keep the procedure going as each row is created, always moving one digit to the right in the real number column as in the illustration (yellow highlight). No matter how long rows are added to the table, X will differ from every real number entry by at least one digit. X then is not in the real number column and never will be no matter how many rows are added to the table. 

Real Number	X
0.2734447…	0.7…
0.3335894…	0.74…
0.4664479…	0.749…
0.5552887…	0.7491…
…	…

That contradicts the assertion that all elements of R,  {x ∊ R | 0 < x < 1} can be enumerated. The contradiction invalidates the assumption which proves the theorem, X is a straggler.