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We show that there exists a non-weakly compact, closed, bounded, convex subset W of the Banach space of convergent sequences (c,‖⋅‖∞), such that every nonexpansive mapping T:W⟶W has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett. This is also the first example of a non-weakly compact, closed, bounded, convex subset W of a Banach space X...