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sequence of integer parts of sequence

Source: Romanian District Olympiad 2015, Grade XI, Problem 4

September 26, 2018
Sequencesreal analysisepsilon-deltacontestromania

Problem Statement

Let (xn)n≥1 \left( x_n\right)_{n\ge 1} be a sequence of real numbers of the interval [1,∞). [1,\infty) . Suppose that the sequence ([xnk])n≥1 \left( \left[ x_n^k\right]\right)_{n\ge 1} is convergent for all natural numbers k. k. Prove that (xn)n≥1 \left( x_n\right)_{n\ge 1} is convergent.
Here, [β] [\beta ] means the greatest integer smaller than β. \beta .
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