Let $c$ be some fixed real number. Let ($a_n$)$_{n=1}^{\infty}$ be a sequence such that for every natural number $n$, $a_n\in\left(c-\frac1 n, c + \frac{1}n\right)$. Prove or disprove: ($a_n$)$_{n=1}^{\infty}$ converges to $c$.
I have seen similar proofs using neighborhoods. Is there a way to prove this without using that this set is a neighborhood of $c$? Is there a way to prove this using just the delta-epsilon definition of continuity?