Am besten von rechts nach links:
\( \frac{1}{m} + \sum \limits_{i=1}^m (p_i - \frac{1}{m} )^2 \)
\( = \frac{1}{m} + \sum \limits_{i=1}^m (p_i^2 -2p_i \frac{1}{m} + (\frac{1}{m} )^2 ) \)
\( = \frac{1}{m} + \sum \limits_{i=1}^m p_i^2 -2 \sum \limits_{i=1}^mp_i \frac{1}{m} + \sum \limits_{i=1}^m (\frac{1}{m} )^2 \)
\( = \frac{1}{m} + \sum \limits_{i=1}^m p_i^2 -\frac{2}{m} \sum \limits_{i=1}^mp_i + m \cdot (\frac{1}{m} )^2 \)
Die Summe der pi ist gleich 1, also
\( = \frac{1}{m} + \sum \limits_{i=1}^m p_i^2 -\frac{2}{m} +\frac{1}{m} = \sum \limits_{i=1}^m p_i^2 \) q.e.d.