Prove that the sequence Yn sinnpi3 has a cluster point at c

Prove that the sequence Y_n = sin(npi/3) has a cluster point at c = squareroot 3/2

Solution

A number x is called a limit point (cluster point, accumulation point) of a set of real numbers A if > 0 ,

( x - , x + ) contains infinitely many points of A.

The sequence is given by y_n = sin( n/3) One can see that for all n one has y_6n = 0, y_6n+1 = (3)/2 , y_6n+2 = (3)/2 , y_6n+3 = 0 , y_6n+4 = (3)/2 , y_6n+5 = (3)/2.

Thus 0 , (3)/2 are the cluster points of the sequence y_n