Proposition 1.17

If $$U$$ is open, $$x \in U$$ and $$x_n \rightarrow x$$, there is some $$N$$ so that $$x_n \in U$$ for all $$n \geq N$$.

Ian_mi
$$ U $$ is open so there exists a ball around $$ x $$, denoted $$ B $$, such that $$ B \subset U $$. $$ x_n \rightarrow x $$ so $$ \exists N \in \mathbb{N} $$ such that $$ x_n \in B \subset U $$ for $$ n \ge N $$ which is what we wished to show.

Ian mi 20:45, February 27, 2011 (UTC)ian_mi