Miklos Schweitzer 1951_2
Source:
October 8, 2008
functionalgebrapolynomialreal analysisreal analysis unsolved
Problem Statement
Denote by a set of sequences S\equal{}\{s_n\}_{n\equal{}1}^{\infty} of real numbers having the following properties:
(i) If S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}, then S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H};
(ii) If S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H} and T\equal{}\{t_n\}_{n\equal{}1}^{\infty}, then
S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H} and ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H};
(iii) \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}.
A real valued function defined on is called a quasi-limit of if it has the following properties:
If S\equal{}{c,c,\dots,c,\dots}, then f(S)\equal{}c;
If , then ;
f(S\plus{}T)\equal{}f(S)\plus{}f(T);
f(ST)\equal{}f(S)f(T),
f(S')\equal{}f(S)
Prove that for every , the quasi-limit is an accumulation point of .