  Let K be a complete discretely valued field with residue field k. If char(K)
= 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an
integer N depending on d such that the u-invariant of any function field in one
variable over K is bounded by N. The method of proof is via introducing the
notion of uniform boundedness for the p-torsion of the Brauer group of a field
and relating the uniform boundedness of the 2-torsion of the Brauer group to
finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group
of function fields in one variable over K are uniformly bounded.
