Great! To walk through a proof involving the definition of a sequence using the epsilon definition, let’s first summarize what this means.

Epsilon-Definition of the Limit of a Sequence:

A sequence $(a_n)$ has the limit $L$ (i.e., $\lim_{n \to \infty} a_n = L$) if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $n \geq N$,
$|a_n - L| < \epsilon.$ In other words, beyond a certain index $N$, the terms $a_n$ are arbitrarily close to the limit $L$.

---

Could you specify the exact sequence you'd like me to prove the limit of using this epsilon-definition? Or should I walk you through a general example like the sequence $a_n = \frac{1}{n}$ converging to 0?

Let me know if you'd like a more specific sequence to be tackled!