What does it mean when we say a sequence is ordered? From wikipedia, we have the definition of a sequence to be
"a sequence is an ordered collection of objects in which repetitions are allowed."
But I don't understand the meaning of "ordered".
For example, do we consider the set of real numbers $\{\pi,2,8,9,5,10,3,4,1,\sqrt{2}\}$ as a sequence?
I would appreciate any help or hint.
Thank you.
 A: The set you exhibit is not a sequence; it is just a set. This set could as well be exhibited as $\{1,\sqrt{2},2,3,\pi,4,5,8,9,10\}$. Maybe these are the numbers you are specially fond of, and you have written them down in the random order they came to your mind. If, however, these numbers are readings of some instrument on successive days you would enclose this list of readings in parentheses instead of braces, like so: $(\pi,2,8,9,5,10,3,4,1,\sqrt{2})$, because it is not only important whether $5$ belongs to this list or not, but also where exactly in this list we see a $5$.
A finite sequence  of length $n\geq1$ is a data structure
$${\bf a}=(a_1,a_2,\ldots, a_n)\tag{1}$$
whereby the $a_i$ can be any objects. Usually all $a_i$ $(1\leq i\leq n)$ belong to some common ground set $X$, say, the real numbers. Mathematically speaking, such an ${\bf a}$ is the same as a function
$$f:\ \{1,2,\ldots,n\}\to X,\qquad i\mapsto a_i\ .$$
Since the domain $[n]:=\{1,2,\ldots,n\}$ of this function is a "standard set" and well known to all concerned this domain is not exhibited expressis verbis in the notation $(1)$; so much the more as the idea of "function" is not felt as an "operation" here, but just as a tool to organize the list $(1)$.
An infinite sequence or sequence, for short, is a function
$$f:\ {\mathbb N}\to X,\qquad k\mapsto x_k\ .$$
Here the idea of a function is much more at stake as before, and our intuition doesn't see an "infinite list" $(x_1,x_2,\ldots)$. Instead we look at the individual $x_k$ as they are produced by $f$ one for one, and we start thinking what the fate of the $x_k$ will be when $k$ is very very large.
A: Consider a nonempty set $X$. Formally, you can define a (finite) sequence of elements of $X$ with length $n \in \mathbb{N}$ as a map $s :\{1,...,n\} \rightarrow X$. Remember that a map $f$ from a set $A$ into a set $B$ is just a special subset of the cartesian product of $A$ and $B$.
$$f \subseteq A \times B, \quad \quad f = \{(a,f(a)) \vert a \in A \} $$
As an example, look at $X = \{a,b,c\}$. Formally, the sequence $s$ whose first element is $a$, second element is $c$, third element is $b$ and fourth element is $a$ again, is a map $$ s: \{1,2,3,4\} \rightarrow X, $$
$$ 1 \mapsto a $$
$$ 2 \mapsto c $$
$$ 3 \mapsto b $$
$$ 4 \mapsto a. $$
If you want to see the sequence as a set, you can view it as 
$$ s = \{(1,a),(2,c),(3,b),(4,a)\}. $$
If you want to define infinite sequences, you just have to take $\mathbb{N}$ as a domain for $s$.
