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A graph is nonrepetitively $k$-choosable if given lists of at least $k$ colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that every graph with maximum degree $\\Delta$ is $c\\Delta^2$-choosable, for some constant $c$. We prove this result with $c=1$ (ignoring lower order terms). 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