I'm going over the proof of the theorem stating that "In a metric space, compactness impliess sequential compactness". I'm very likely confusing myself. I have the following proposition:
$\forall x\in X$, $\exists \epsilon(x)$ such that $B(x;\epsilon(x))$ contains only finitely many terms from the sequence $\{y_j\}$.
Now I have already proved that this is not the case (since this is true then the covering I found would actually miss infinitely many terms from the sequence). Then I tried to negate this proposition. Then I confused myself. Should it be like:
$\exists x\in X $ such that $!\exists \epsilon$ satisfying that $B(x;\epsilon)$ contains only finitely many terms.
But if this is true, what about a ball contains no term from the sequence?