= Solution
<EM likelihood monotonicity> makes EM an ascent method, not a global-optimization theorem. If the <log-likelihood> is bounded above along the iterates, its nondecreasing values converge. That alone does not show that the parameters converge, nor that any limiting parameter is a global <maximum-likelihood estimator>. Under additional regularity, accumulation points can be stationary; local maxima, other stationary points or flat sets can depend on initialization. Multiple starting values and comparison of the achieved observed likelihoods are therefore useful.
The upper-bound hypothesis is substantive. In a <finite mixture model> with a freely shrinking normal-component <variance>, placing one component at an observed value can make the <likelihood> unbounded while other components cover the remaining observations. In that situation monotonicity provides no finite maximum to converge to. EM may also improve very slowly, and a small change in <likelihood> alone is not proof of an accurate global solution. The guarantee in part (b) is nondecrease, not strict increase at every iteration.
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