 Suppose that \(g(j) \uarrow\). Then \(h(j,j) \neq 0\) by definition of \(g(x)\). Since \(h(x,y)\) is the characteristic function of \(\HP\) (and hence total), the only other possibility is that \(h(j,j) = 1\) which in turn implies that \(\phi_j(j) \darrow\). But then since \(g(x) \simeq \phi_j(x)\), \(g(j) \darrow\), a contradiction. □ 