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?
Best Answer
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 $I = \mathbb{N}$.