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sequence accumulating everywhere in
[−1, 1]∗
pahio†
2013-03-22 3:30:59
We want to show that if k is an irrational number, then any real number of
the interval [−1, 1] is an accumulation point of the sequence
sin 2kπ, sin 4kπ, sin 6kπ, . . .
(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 , P 1 , P 2 , . . .
(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
2π
< ε
n
and divide 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 P0 P1 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 Pµ Pν 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
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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].
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