Sturm sequences or law of inertia of quadratic forms?

In obtaining the number of eigenvalues greater than a given constant λ₀ of the eigenvalue problem A]{x} = λB]{x} with A] and B] real, symmetric, and B] positive definite, one usually refers to the Sturm sequence established by the leading principal minors of A–λ₀B], the proof of which is given basically when the associated special eigenvalue problem is tridiagonal. In this work, using the law of inertia of quadratic forms, it is shown that the number of eigenvalues of A]{x}] = λB]{x} greater than a given constant λ₀ (not an eigenvalue) is the number of positive entries of the diagonal matrix d] in the identity A–λ₀B] = u]^Td]u] where u] is the upper triangular matrix associated with Crout-Banachievicz type decomposition of A –λ₀B], without the help of the separation theorem and the Sturm sequence.