Abstract
Proof. Suppose the Ai satisfy (1) for every f E Ep . We leave it to the reader to show that Im (Aj) is bounded; this is analogous to the argument below, but simpler. Supposing this, we show that the Ai can be partitioned into a finite union of separated sequences. We need only show that the number of Aj with real part lying in the interval [,B,,B + 1] is bounded. Supposing that this is false, we shall construct an f E Ep for which (1) fails: Choose A > 1 so that IIm(Aj)l 1 for all z E C
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