  We introduce an ordinal index which measures the complexity of a weakly null
sequence, and show that a construction due to J. Schreier can be iterated to
produce for each alpha < omega_1, a weakly null sequence (x^{alpha}_n)_n in
C(omega^{omega^{alpha}})) with complexity alpha. As in the Schreier example
each of these is a sequence of indicator functions which is a suppression-1
unconditional basic sequence. These sequences are used to construct
Tsirelson-like spaces of large index. We also show that this new ordinal index
is related to the Lavrentiev index of a Baire-1 function and use the index to
sharpen some results of Alspach and Odell on averaging weakly null sequences.
