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-2⌊log2n⌋2⌊log2n⌋,n-2⌊log2n⌋+12⌊log2n⌋]⊆[0,1].
Proof.
Note that this is equivalent (http://planetmath.org/Equivalent) to showing that, for every positive integer n,
n-2⌊log2n⌋≥0 and n-2⌊log2n⌋+1≤2⌊log2n⌋. This in turn is equivalent to showing that, for every positive integer n, 2⌊log2n⌋≤n and n+1≤2⌊log2n⌋+1.
The first inequality is easy to prove: For every positive integer n, 2⌊log2n⌋≤2log2n=n.
Now for the second inequality. Let n be a positive integer. Let k be the unique positive integer such that
2k-1≤n≤2k-1. Then n+1≤2k=2k-1+1=2⌊k-1⌋+1=2⌊log22k-1⌋+1≤2⌊log2n⌋+1. ∎
Title | regarding the sets An from the traveling hump sequence |
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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 |