Homework Help: Limit point.

1. Feb 7, 2012

cragar

1. The problem statement, all variables and given/known data
a)Determine at least three limit points for the set {sin(n): n a positive integer}
b)How many limit points does the set {sin(n): n a positive integer} have?
3. The attempt at a solution
For a it seems that it wouldn't have a limit point because sin(n) would not converge to anything. On the other hand maybe its obvious and maybe 3 limit points would be
sin(1), sin(2), sin(3) , now that I think about it, it seems like a tricky question.
for part b, if sin(n) has 3 limit points then it seems like the set would have an infinite amount of points.

2. Feb 7, 2012

aeroplane

From http://en.wikipedia.org/wiki/Limit_point we have that x is a limit point of A={sin(n): n a positive integer} if every neighborhood of x has another element of A different from x. So if you can show that you can approximate x as closely as you wish with elements taken from A, then you have a limit point there. I think three natural points to try to show are limit points are 0,1,-1.

3. Feb 8, 2012

cragar

so really and point in [-1,1] should be a limit point because i can get as close as I want to any point in there with the sin(n), can points in my set {sin(n)} where n is a natural number.
be a limit point?

Last edited: Feb 8, 2012