 Suppose that \(g(j) \darrow\). Then \(h(j,j) = 0\) by definition of \(g(x)\). Since \(h(i,x)\) is the characteristic function of \(\HP\), this means \(\phi_j(j) \uarrow\). But then since \(g(x) \simeq \phi_j(x)\), \(g(j) \uarrow\), a contradiction.