im having a little doubt about convergence in metric spaces. So we know that for an arbitrary topological space a sequence of points converges to $x$ if corresponding to each neighborhood $U$ of $x$ there exists a natural number $N$ such that $x_n \in U$ for all $n \geq N$.
Obviously this applies to metric spaces. My question is that if we are in metric spaces to prove that a sequence converges to a points $x$ it is enough to prove that for any Ball of radius $\epsilon >0$, $B_\epsilon$ there exists a natural number $N$ such that $x_n \in B_\epsilon$ for all $n \geq N$. I guess this can be generalized to basis elements, but yeah thats my question. This seems to be true to me but i am not really convinced, so any help is appreciated.
Hint: if $U$ is any open set containing $x$ then there exists $\epsilon >0$ such that $B(x,\epsilon) \subset U$.