Can subsequences be finite? Say I'm given the sequence $\{a_1,a_2,a_3,\dots\}$. Does a subsequence have to be infinite? Or can it be finite too? For example, is $\{a_1,a_2,a_3\}$ a subsequence? 
 A: Yes the subsequence must be infinite. Any subsequence is itself a sequence, and a sequence is basically a function from the naturals to the reals.
A: Usually, this is the definition of subsequence.
Definition. Let $\{p_n\}_n$ be a sequence (in some set), and let $n_1<n_2<n_3<\ldots$ be a strictly increasing sequence of positive integers. Then the sequence $\{p_{n_j}\}_j$ is called a subsequence of $\{p_n\}_n$.
It follows that $\{p_1,p_2,p_3\}$ is not a subsequence of $\{p_n\}_n$, since $n_1=1$, $n_2=2$, $n_3=3$, but what is $n_4$? This is the common feeling for mathematical analysts.
However, we must be careful, since it all boils down to the very definition of sequence. In analysis, a sequence is a function $\mathbb{N} \to X$, or, more generally, a function $N \to X$, where $N \subset \mathbb{N}$ is such that $N$ contains all sufficiently large integers.
In other disciplines, it might be useful to call sequence any function defined on a subset of $\mathbb{N}$, even a finite one.
A: A sequence in a set $X$ is a function $f:\mathbb{N}\rightarrow X$. Let $g:\mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing sequence, then the composition of $f$ and $g$, is called a subsequence of $f$. Thus the domain of a subsequence is always infinite, but the range can certainly be finite.
As an example consider the sequence $f(n):=n$ even $n$ and $f(n):=1$ odd $n$.  Let $g(n):=2n+1$. The range of $f\circ g$ is equal to $\{1\}$, which is finite.
A: In my experience, the term subsequence has always referred to an infinite subsequence. I believe this is standard. If you want to consider finite subsequences, you should say finite subsequence, or if you want to consider subsequences that may or may not be finite, you can say possibly finite subsequence.
A: A sequence (real, sets, etc...) is simply a list. In mathematics, unless specified, a finite list (i.e. sequence) is not denoted to be a sequence. A sequence MUST continue unimpeded. 
By definition, a sequence of real numbers is a mapping from N -> R. The nature of this definition demonstrates that a finite sequence cannot exist under our current consideration. 
