Finite Cauchy Sequence

Question

How do you prove that given a sequence that is finite, the sequence always converges? Or, must the sequence be cauchy for this finite sequence to converge? How can a cauchy sequence be finite? If a cauchy sequence is finite, won't all the elements in the sequence be arbitrarily close?

Answer 1

Since the given sequence is finite, there is a term say $b$ which repeats infintely many times i.e there exists an $n_0$ such that for all $n \ge n_0$, we have $a_n=b$. And the consequences follow

Answer 2

Let $\{ a_k \}_{k=1}^{n}$ be a finite sequence. This sequence converges to $a_n$, because for all $k \geq n \in \mathbb{N}$, $|a_k - a_n| = |a_n - a_n| = 0 < \epsilon$ for any $\epsilon > 0$.

It's not a very interesting case, but a good problem for understanding the definition of convergence.