Sequences in uniformly convex Banach spaces satisfying the norm inequality ||v_{n+m}|| <= ||v_n + v_m|| have a convergent limit v_n/n.
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Fekete's lemma in Banach spaces
Sequences in uniformly convex Banach spaces satisfying the norm inequality ||v_{n+m}|| <= ||v_n + v_m|| have a convergent limit v_n/n.