sequence accumulating everywhere in [-1,\tmspace+.1667em1]

We want to show that if k is an irrational number, then any real number of the interval[-1, 1]  is an accumulation pointMathworldPlanetmathPlanetmath of the sequence

sin2kπ,sin4kπ,sin6kπ, (1)

In other words, the real numbers (1) come arbitrarily close to every number of the interval.

Proof.  Set on the perimeter of the unit circle, starting e.g. from the point  (1, 0),  anticlockwise the points

P0,P1,P2, (2)

with successive arc-distances 2kπ.  Since k is irrational, 2kπ and the length 2π of the perimeter are incommensurable.  Therefore no two of the points Pi coincide, whence we have an infinite sequence (2) of distinct points.  We can see that these points form an everywhere dense set on the perimeter, i.e. that an arbitrarily short arc contains always points of (2).

Let then ε be an arbitrary positive number.  Choose an integer n such that


and the perimeter of the unit circle, starting from the point  (1, 0),  into n equal arcs.  Each of the points P1,P2, falls into one of these arcs, because the arcs 2π/n and P0P1 are incommensurable.  Thus, among the n+1 first points P1,P2,,Pn+1 there must be at least two ones belonging to a same arc.  Let Pμ and Pν (μ<ν) belong to the same arc.  Then the length l of the arc PμPν is less than ε.  Starting from Pμ one comes to Pν by moving on the perimeter ν-μ times in succession arcs with length 2kπ (when one has possibly to go around the perimeter several times).  Repeating that procedure, starting from the point Pν, one comes to the point  Pν+(ν-μ)=P2ν-μ, and furthermore to P3ν-2μ, to P4ν-3μ, and so on.

The points

Pμ,Pν,P2ν-μ,P3ν-2μ, (3)

form on the perimeter a sequence of equidistant points, a subsequence of (2).  Since the arc-distance of successive points of (3) equals to  l<ε, whence it is evident that any arc with length at least ε contains at least one of the points (3).  Consequently, the points (2) are everywhere dense on the perimeter of the unit circle.  Thus the same concerns their projections ( on the y-axis, i.e. the sines (1) on the interval  [-1, 1].

Title sequence accumulating everywhere in [-1,\tmspace+.1667em1]
Canonical name SequenceAccumulatingEverywhereIn11
Date of creation 2013-03-22 19:13:39
Last modified on 2013-03-22 19:13:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 54A05
Related topic LissajousCurves