Is the limit of a finite sequence simply its last term? I haven't dealt with convergence of finite sequences yet and my textbook doesn't say much about this. Using the definition of convergence, I was able to show that the limit of a finite sequence is its last term. I don't know if this is correct. Can you please let me know?
Let $\{a_n \}_{n \in N}$ be a finite sequence where $N \subset \mathbb{N}$. Denote the number of elements in $N$ by $\bar{N}$. Now we know that for all $\epsilon > 0, \ \exists \ \bar{N} \in \mathbb{N}$ such that if $ n \geq \bar{N}, \ \mid a_\bar{N}-a_n \mid<\epsilon$. But this is the definition of a sequence that converges to $a_\bar{N}$. So does this mean that the limit of a finite sequence is simply the last element of the sequence?
 A: Technically yes, I think, but it is not too interesting to consider convergence of a sequence that terminates after finitely many terms.
If $a_1,...,a_{n_0}$ is a finite sequence indexed by $I = \{1,...,n_0\}$ and $x = a_{n_0}$, then the criterion for convergence $a_n \to x$ is automatically satisfied: $$(\forall \varepsilon > 0)(\exists N_0 \in I)(n \geqslant N_0, \, n \in I \,\, \Longrightarrow  \,\,|a_n-x|<\varepsilon).$$ No matter what $\varepsilon$ you pick setting $N_0 = n_0$ will work.
Typically the index set is $I = \mathbb{N}$.
A: The limit is usually only defined for an infinite sequence, not a finite sequence.  
The usual definition:

For all $\epsilon > 0$ there is $N$ such that for every $n > N$, $|a_n - L| < \epsilon$

assumes $a_n$ to be defined for every $n > N$, which is not the case for a finite sequence.
If you modify it to say "... for every $n > N$ for which $a_n$ is defined, ...", then the limit of a finite sequence could be anything at all.  There's no particular reason for it to be the last member of the sequence: $L = $"a unicorn"
 works just as well if $N$ is greater than the index of the last member of the sequence.
