Folks,

Are the following subsets closed?

1) $\displaystyle S=(0,1]; in X=R$

2) $\displaystyle S=\{x=(x_1,x_2) \in R^2 : x_1^2+x_2^2 \le 1 \} in X=R$

For 1) $\displaystyle S=(0,1]; in X=R$ is not closed

for if $\displaystyle S_n$ is a convergent sequence with $\displaystyle 0 < S_n \le 1 \forall n \in N$

$\displaystyle S-N \le 1 \forall n \implies Lim S_n \le Lim 1 =1 as n \rightarrow \infty$

I dont know if this is right or how to continue.....?