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A bounded real sequence {\displaystyle } is said to be almost convergent to L {\displaystyle L} if each Banach limit assignsthe same value L {\displaystyle L} to the sequence {\displaystyle }.
Lorentz proved that {\displaystyle } is almost convergent if and only if
uniformly in n {\displaystyle n}.
The above limit can be rewritten in detail as
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