S (for student) is a function that creates a sequence of real numbers between 0 and 1 noninclusive. S is defined such that S(i, j), i and j both positive integers, is the jth digit of the ith real number in the sequence.

M is the matrix represented by S, it is the infinite matrix whose i'th row is the sequence .

If T (for "teacher") is a function from positive integers to digits 0-9, The sequence represented by T is defined as the sequence .

We say that T beats S if the sequence T represents is not a row in the matrix S represents. Otherwise S beats T.

Give a T function that can beat any possible S, or in other words give a T function that cannot be beaten by an arbitrary perfect S function.

Additionally, explain why this T function shows that R(0,1) is not equinumerous with N