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8

Part of 1956 Miklós Schweitzer

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Miklós Schweitzer 1956- Problem 8

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10/11/2015
8. Let (an)n=1∞(a_n)_{n=1}^{\infty} be a sequence of positive numbers and suppose that ∑n=1∞an2\sum_{n=1}^{\infty} a_n^2 is divergent. Let further 0<ϵ<120<\epsilon<\frac{1}{2}. Show that there exists a sequence (bn)n=1∞(b_n)_{n=1}^{\infty} of positive numbers such that ∑n=1∞bn2\sum_{n=1}^{\infty}b_n^2 is convergent and
∑n=1Nanbn>(∑n=1Nan2)12−ϵ\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}
for every positive integer NN. (S. 8)
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