On the u-invariant of function fields of curves over complete discretely valued fields

Let K be a complete discretely valued field with residue field κ. If char(K)=0, char(κ)=2 and [κ:κ2]=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 the finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over K is uniformly bounded.