MathDB
decreasing function

Source: flanders '90 (could be posted before?)

August 9, 2004
functionlimitcalculuscalculus computations

Problem Statement

Let f:R0+→R0+f:\mathbb{R}^+_0 \rightarrow \mathbb{R}^+_0 be a strictly decreasing function. (a) Be ana_n a sequence of strictly positive reals so that ∀k∈N0:k⋅f(ak)≥(k+1)⋅f(ak+1)\forall k \in \mathbb{N}_0:k\cdot f(a_k)\geq (k+1)\cdot f(a_{k+1}) Prove that ana_n is ascending, that lim⁡k→+∞f(ak)\displaystyle\lim_{k\rightarrow +\infty} f(a_k) = 0and that lim⁡k→+∞ak=+∞\displaystyle\lim_{k\rightarrow +\infty} a_k =+\infty (b) Prove that there exist such a sequence (ana_n) in R0+\mathbb{R}^+_0 if you know lim⁡x→+∞f(x)=0\displaystyle\lim_{x\rightarrow +\infty} f(x)=0.
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