MathDB
CIIM 2018 Problem 6

Source:

March 10, 2019
undergraduatecollege contestsSequencereal analysis

Problem Statement

Let {xn}\{x_n\} be a sequence of real numbers in the interval [0,1)[0,1). Prove that there exists a sequence 1<n1<n2<n3<⋯1 < n_1 < n_2 < n_3 < \cdots of positive integers such that the following limit exists lim⁡i,j→∞xni+nj.\lim_{i,j \to \infty} x_{n_i+n_j}. That is, there exists a real number LL such that for every ϵ>0,\epsilon > 0, there exists a positive integer NN such that if i,j>Ni,j > N, then ∣xni+nj−L∣<ϵ.|x_{n_i+n_j}-L| < \epsilon.
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