Changes

Obviously, <math>a_1 \ </math> is contained in this interval, and this interval has length <math>w/4+w/4 = w/2 \ </math>.   

Obviously, <math>a_1 \ </math> is contained in this interval, and this interval has length <math>w/4+w/4 = w/2 \ </math>.   

But around <math>a_2 \ </math>, we can construct the interval <math>[a_2-w/8,a_2+w/8] \ </math>, around <math>a_3, [a_3-w/16,a_3+w/16] \ </math>, and in general, around <math>a_k \ </math>, we can construct <math>[a_k-w2^{k+1},a_k+w2^{k+1}] \ </math>.   

But around <math>a_2 \ </math>, we can construct the interval <math>[a_2-w/8,a_2+w/8] \ </math>, around <math>a_3, [a_3-w/16,a_3+w/16] \ </math>, and in general, around <math>a_k \ </math>, we can construct <math>[a_k-w/2^{k+1},a_k+w/2^{k+1}] \ </math>.   

The sum of all these lengths will be <math>w/2 + w/4 + w/8 + w/16 + ... = w \ </math>, but remember, <math>w \ </math> was chosen to be arbitrarily small, and so a countable set can be contained in a set of arbitrarily small measure.

The sum of all these lengths will be <math>w/2 + w/4 + w/8 + w/16 + ... = w \ </math>, but remember, <math>w \ </math> was chosen to be arbitrarily small, and so a countable set can be contained in a set of arbitrarily small measure.

Since the real numbers, or any set of positive measure, cannot, they must not be countable.

Since the real numbers, or any set of positive measure, cannot, they must not be countable.