Convergence in topological space

Question

I need to show that a convergent sequence in a metric space $(X,d)$ converges in a topological space $(X,t)$ where $t$ is the topology generated by $d$ on $E$.

By definition, for every $\epsilon >0, \exists N: \forall n>N: d(x,x_n)<\epsilon$ with $x\in E$. So, eventually the sequence is in an arbitrary open ball centered at $x$ with radius $\epsilon$. As far as I understand, the exercise is asking me to show that for an arbitrary open set in $(E,d)$ which contains $x$, the sequence is eventually in this set. Am I right?

How can I do that?

Thanks!

Answer 1

Yes, you are right.

Let $U$ be an open set which contains $x$. By definition, there exists $r > 0$ such that the open ball $B(x,r) \subseteq U$. Can you finish?