TITLE: Finite Cauchy Sequence
QUESTION [1 upvotes]: 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?

REPLY [2 votes]: 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.
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