In this work we investigate bounded Lukasiewicz logics}, characterised as the intersection of the k-valued Lukasiewicz logics for k = 2, ..., n (n ≥ 2). These logics formalise a generalisation of Ulam’s game with applications in Information Theory. Here we provide an analytic proof calculus GŁ B _n for each bounded Lukasiewicz logic, obtained by adding a single rule to GŁ, a hypersequent calculus for Lukasiewicz infinite-valued logic. We give a first cut-elimination proof for GL with (suitable forms of) cut rules. We then prove completeness for GŁ B _n with cut and show that cut can also be eliminated in this case.