Proposition 1.18

If $$C$$ is closed, $$x_n \in C$$ for each $$n$$ and $$x_n \rightarrow x$$, then $$x \in C$$.

Ian_mi
Suppose $$ x \in C^c $$. $$ C^c $$ is open so, by proposition 1.17, $$ \{x_n\} $$ is eventually in $$ C^c $$. This contradicts the assumption that $$ \{x_n\} \subseteq C $$. Thus $$ x \in C $$.

Ian mi 20:39, February 27, 2011 (UTC)ian_mi

'''Cool! --Steven.clontz 02:43, March 11, 2011 (UTC)'''