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We show that the set of numbers with bounded L\"uroth expansions (or boundedL\"uroth series) is winning and strong winning. From either winning property,it immediately follows that the set is dense, has full Hausdorff dimension, andsatisfies a countable intersection property. Our result matches the well-knownanalogous result for bounded continued fraction expansions or, equivalently,badly approximable numbers.  We note that L\"uroth expansions have a countably infinite Markov partition,which leads to the notion of infinite distortion (in the sense of Markovpartitions).

Bounded L\"uroth expansions: applying Schmidt games where infinite distortion exists