Consider the sequence \(w_{0}, w_{1}, \ldots\),where \(w_{0}=g(\boldsymbol{\theta})\) is a non-degenerate initial guess and \(w_{t}(\boldsymbol{\theta}) \propto w_{t-1}(\boldsymbol{\theta}) g(\tau \mid \boldsymbol{\theta}), t>1\). We assume that \(g(\tau \mid \boldsymbol{\theta})\) is not the constant function (with respect to \(\theta\) ) and that the maximum likelihood value

\[ g(\tau \mid \widehat{\boldsymbol{\theta}})=\max _{\boldsymbol{\theta}} g(\tau \mid \boldsymbol{\theta})<\infty \]

exists (is bounded). Let

\[ l_{t}:=\int g(\tau \mid \boldsymbol{\theta}) w_{t}(\boldsymbol{\theta}) \mathrm{d} \boldsymbol{\theta} \]

Show that \(\left\{l_{t}\right\}\) is a strictly increasing and bounded sequence. Hence, conclude that its limit is \(g(\tau \mid \widehat{\boldsymbol{\theta}})\).