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sequences of real numbers

Source: OBM 2017

March 20, 2023
real analysisSequencesHigh school olympiad

Problem Statement

Let (an)n≥1(a_n)_{n\geq 1} be a sequence of positive real numbers in which lim⁡n→∞an=0\lim_{n\to\infty} a_n = 0 such that there is a constant c>0c >0 so that for all n≥1n \geq 1, ∣an+1−an∣≤c⋅an2|a_{n+1}-a_n| \leq c\cdot a_n^2. Show that exists d>0d>0 with nan≥d,∀n≥1na_n \geq d, \forall n \geq 1.
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