  A number is normal in base $b$ if, in its base $b$ expansion, all blocks of
digits of equal length have the same asymptotic frequency. The rate at which a
number approaches normality is quantified by the classical notion of
discrepancy, which indicates how far the scaling of the number by powers of $b$
is from being equidistributed modulo 1. This rate is known as the discrepancy
of a normal number. The Champernowne constant $c_{10} =
0.12345678910111213141516\ldots$ is the most well-known example of a normal
number. In 1986, Schiffer provided the discrepancy of numbers in a family that
includes the Champernowne constant. His proof relies on exponential sums. Here,
we present a discrete and elementary proof specifically for the discrepancy of
the Champernowne constant.
