To prove that every eventually constant sequence is convergent, let's go through the formal steps.

Definitions:

1. A sequence $\{a_n\}$ is called eventually constant if there exists some index $N$ such that for all $n \geq N$, $a_n = c$, where $c$ is a constant.

2. A sequence $\{a_n\}$ is said to converge to a limit $L$ if for every $\epsilon > 0$, there exists an integer $N_\epsilon$ such that for all $n \geq N_\epsilon$, $|a_n - L| < \epsilon$.

Proof:

Let $\{a_n\}$ be an eventually constant sequence. By the definition of "eventually constant," there exists an index $N$ such that: $a_n = c \quad \text{for all} \quad n \geq N,$ where $c$ is a constant.

We will show that $\{a_n\}$ converges to $c$.

Step 1: Choose $\epsilon > 0$.

We need to find an index $N_\epsilon$ such that for all $n \geq N_\epsilon$, $|a_n - c| < \epsilon.$

Step 2: Find $N_\epsilon$.

Since $a_n = c$ for all $n \geq N$, we can choose $N_\epsilon = N$. For any $n \geq N$, $|a_n - c| = |c - c| = 0 < \epsilon.$ Thus, the condition $|a_n - c| < \epsilon$ holds for all $n \geq N_\epsilon$.

Step 3: Conclusion.

Since for any $\epsilon > 0$, there exists an index $N_\epsilon = N$ such that $|a_n - c| < \epsilon$ for all $n \geq N_\epsilon$, we conclude that the sequence $\{a_n\}$ converges to $c$.

Therefore, every eventually constant sequence is convergent.

---

Do you have any questions, or would you like further details?

Here are 5 related questions you might find interesting:

1. Can a non-constant sequence converge to a constant?
2. How does this proof compare to proving the convergence of a Cauchy sequence?
3. What is the relationship between boundedness and convergence in sequences?
4. Can you provide an example of a sequence that is not eventually constant but converges?
5. How does this result apply to sequences in metric spaces?

Tip: When proving convergence, always rely on the $\epsilon$-definition to ensure rigor in your arguments.