Cantor tells us that infinities come in "sizes" or cardinalities.
Some infinities are "larger" than others.
The smallest infinite set comprises the counting numbers, 1 2 3 4 5 ...
and any other set that can be put in a 1-1 correspondence with them.
Such infinities are called "countable."
The rational numbers are also countable, but the real numbers are not.
Occasionally someone will attempt to pair the counting number with the reals.
One such scheme is shown below, and the argument goes as follows:
The left column is the endless list of integers in numerical order
The right column contains a decimal fraction formed by reversing the digits and placing a decimal point in front.
Since the left hand list proceeds without limit it eventually contains every possible sequence of digits.
Then the right hand list also catches every sequence of digits and thus represents the real numbers less than unity.
Numbers greater than unity can be constructed by sets with a shift in the position of the decimal point.
The union of two or more sets of equal cardinality has the same cardinality as its component sets.
This correspondence is sufficient to prove the two sets have the same cardinality.
The argument is obviously flawed, and the question is to expose the flaw.
Here are the lists:
Integers Decimal fractions