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RMM and Analysis? Why not bro!

Source: RMM Shortlist 2021 A4

September 18, 2023

analysisRMM ShortlistSequencesConvergencealgebra

Problem Statement

Let

f: \mathbb{R} \to \mathbb{R}

be a non-decreasing function such that

f(y) - f(x) < y - x

for all real numbers

x

and

y > x

. The sequence

u_1,u_2,\ldots

of real numbers is such that

u_{n+2} = f(u_{n+1}) - f(u_n)

for all

n\geq 1

. Prove that for any

\varepsilon > 0

there exists a positive integer

N

such that

|u_n| < \varepsilon

for all

n\geq N

.

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