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If ξ is a countable ordinal and (fk) a sequence of real-valued
functions we define the repeated averages of order ξ of (fk). By using a
partition theorem of Nash-Williams for families of finite subsets of positive
integers it is proved that if ξ is a countable ordinal then every sequence
(fk) of real-valued functions has a subsequence (f'k) such that either every
sequence of repeated averages of order ξ of (f'k) converges uniformly to zero
or no sequence of repeated averages of order ξ of (f'k)...
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