MathDB
hard limit

Source: IMC 2000 day 1 problem 6

October 29, 2005
functionlimitintegrationreal analysisreal analysis unsolved

Problem Statement

Let f:R→]0,+∞[f: \mathbb{R}\rightarrow ]0,+\infty[ be an increasing differentiable function with lim⁡x→+∞f(x)=+∞\lim_{x\rightarrow+\infty}f(x)=+\infty and f′f' is bounded, and let F(x)=∫0xf(t)dtF(x)=\int^x_0 f(t) dt. Define the sequence (an)(a_n) recursively by a0=1,an+1=an+1f(an)a_0=1,a_{n+1}=a_n+\frac1{f(a_n)} Define the sequence (bn)(b_n) by bn=F−1(n)b_n=F^{-1}(n). Prove that lim⁡x→+∞(an−bn)=0\lim_{x\rightarrow+\infty}(a_n-b_n)=0.
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