The simplest example I can think of is the interval (0, 1].
Proof that it's not closed: the sequence (1, 1/2, 1/4, 1/8, ...) is entirely inside the interval, but converges on 0, which is outside the interval, therefore etc.
Proof that it's not open: the sequence (2, 3/2, 5/4, 9/8, ...) is entirely outside the interval (i.e. inside the complement), but it converges on 1, which is inside the interval (i.e. outside the complement). Thus the complement of the interval is not closed, therefore etc.
Comments
The simplest example I can think of is the interval (0, 1].
Proof that it's not closed: the sequence (1, 1/2, 1/4, 1/8, ...) is entirely inside the interval, but converges on 0, which is outside the interval, therefore etc.
Proof that it's not open: the sequence (2, 3/2, 5/4, 9/8, ...) is entirely outside the interval (i.e. inside the complement), but it converges on 1, which is inside the interval (i.e. outside the complement). Thus the complement of the interval is not closed, therefore etc.