MathDB
Miklos Schweitzer 1950_3

Source: second part of 1950

October 3, 2008
algebra proposedalgebra

Problem Statement

For any system x1,x2,...,xn x_1,x_2,...,x_n of positive real numbers, let f_1(x_1,x_2,...,x_n) \equal{} x_1, and f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}} for \nu \equal{} 2,3,...,n. Show that for any ϵ>0 \epsilon > 0, a positive integer n0<n0(ϵ) n_0 < n_0(\epsilon) can be found such that for every n>n0 n > n_0 there exists a system x1′,x2′,...,xn′ x_1',x_2',...,x_n' of positive real numbers with x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1 and fν(x1′,x2′,...,xn′)≤ϵ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon for \nu \equal{} 1,2,...,n .
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