The Infimum in the Metric Mahler Measure Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure. Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problemCategories:11R04, 11R09