- 62

- 0

**1. Homework Statement**

Find a sequence whose set of subsequential limits is the interval [0,1].

**2. Homework Equations**

If the sequence does not repeat itself, then any subsequential limit is a cluster point.

**3. The Attempt at a Solution**

I've an idea that [itex]|\sin n|[/itex] is a solution to this, but I'm not sure how to prove this.

What I need to show is that, since the sequence I've chosen is non-repeating, then every point in the interval [0,1] is a cluster point. That is, if I take any number in [0,1] and am given an epsilon>0, can I always find an infinite amount of members in the sequence in an epsilon neighborhood around the number? This seems to intuitively work since [itex]|\sin n| [/itex] is all over the interval [0,1] (though not necessarily touching every point in the interval), but can this argument be made more rigorous? Any suggestions?