regarding the sets An from the traveling hump sequence


In this entry, denotes the floor function.

Following is a proof that, for every positive integer n, [n-2log2n2log2n,n-2log2n+12log2n][0,1].

Proof.

Note that this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent) to showing that, for every positive integer n,

n-2log2n0 and n-2log2n+12log2n. This in turn is equivalent to showing that, for every positive integer n, 2log2nn and n+12log2n+1.

The first inequalityMathworldPlanetmath is easy to prove: For every positive integer n, 2log2n2log2n=n.

Now for the second inequality. Let n be a positive integer. Let k be the unique positive integer such that

2k-1n2k-1. Then n+12k=2k-1+1=2k-1+1=2log22k-1+12log2n+1. ∎

Title regarding the sets An from the traveling hump sequence
Canonical name RegardingTheSetsAnFromTheTravelingHumpSequence
Date of creation 2013-03-22 16:14:28
Last modified on 2013-03-22 16:14:28
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 6
Author Wkbj79 (1863)
Entry type Proof
Classification msc 28A20