Sequences of Real Numbers

In my previous posts, i have written a lot about many topics of the real number system,
today i am going to be writing on a very vital aspect of the real number
"the sequences",this is one topic we can not do without in our study of the real numbers.
Logically we think of the sequences as a "list",it may be a list of sets,numbers,functions
or anything.Once we think of the sequence then we probably think of a list of something.
Now in every list,there is an order of arrangement, in the sense that there is always a
first element,second,third,fourth continuously with out an end.we must note that a finite
list is not called a sequence, a sequence always continues without an interruption.
for a more formal definition you will notice that the natural numbers are playing a key
role here.
The mathematical definition of a sequence is simply :
A function from the set of Natural numbers $\mathbb{N}$ to the set of real numbers
$\mathbb{R}$. i.e $f:\mathbb{N}\rightarrow\mathbb{R}$, thus we can say the sequence is
the function,in the sense that $f(1),f(2),f(3),...,f(n)$.here the function values
$f(1),f(2),f(3),...$ are called the terms of the sequence.A sequence can also be
represented in the form $\{f(n)\}$, understanding that $n$ ranges over all of the natural
numbers.
Mathematically, sequences are usually written with a subscript notation like this.
$f_1,f_2,f_3,...,f_n$ or $\{f_n\}$

Examples of sequences.
Below are some perfect examples of sequences, note that examples of sequences can be generated using the recursive formulas, but here i am only going to list the standard examples.
(1)Arithmetic Progression: The A.P are sequences in which each term is obtained from the
preceding by adding a fixed amount. i.e $a,a+d,a+2d,a+3d,...,a+(n-1)d$ here a is known as
the first term,d is the common difference,the general formula for an A.P is
$x_n=a+(n-1)d$.
(2)Geometric Progression: The G.P is a sequence i which each term is obtained from the
preceding by the multiplication with a fixed amount. i.e
$a,ar,ar^2,ar^3,ar^4,...,ar^{n-1}$. here a is the first term, while r is called the
common difference.in general the G.P has a formula of $x_n=ar^{n-1}$.
(3)Sequence of Partial Sums:The sequence of partial sums is obtained by simply adding the
terms of the sequence to a new one.for example
\begin{align*}
s_1=x_1\\
s_2=x_1+x_2\\
s_3=x_1+x_2+x_3\\
s_4=x_1+x_2+x_3+x_4
\end{align*}
this process can be described by a recursion formula $s_n=s_{n-1}+x_n$, Here a new
sequence is called the sequence of partial sums of the old sequence $\{x_n\}$

Countable Sets
A non-empty set  $X$ of real numbers is said to be countable if there is a sequence of
real numbers whose range is the set $X$.
In other words a set is countable if there exists a one-one correspondence function.
Now in the language of this definition we can say that.
(4)No interval of the real numbers is countable, be it the open,closed,half open or half
closed.
(5) Conventionally, we can say that the empty set $\phi$ is countable.
Now we can give a definition of an uncountable set,
An uncountable set is a set that cannot be represented in one-one correspondence function.