epsilon-delta proofs 2

Do you remember that real numbers consist of rational numbers, irrational numbers, and 0? And all rational numbers have fractional expressions $p/q$, in which $p$ is an integer, and $q$ is also an integer not zero. In calculations using fractions, the value of $p/q$ is a positive or negative integer, 0, or a terminating or recurring decimal.

As explained in the previous post, integers or terminating decimals are equal to a recurring decimal. As irrational numbers originally are numbers recurring infinitely, it follows that all real numbers except 0 also recurre infinitely.

Since it is possible to construct a real number sequence by analyzing the number having infinite digits, all real numbers can be expressed a real number sequence. For instance, when $a=2.4$, its sequence $a_n$ is as follows:

$a_0=2.3$
$a_1=2.39$
$a_2=2.399$
$a_3=2.3999$
$\cdots \cdots$

If we make $n$ infinite, then $a_{\infty }=a=2.4$. But if $n$ is finite, the more you increase $n$, the smaller the difference $|a_n-a|$ becomes. I will give the definition of $a_n$ approches $a$.

[Def-1] $a_n$ approches $a$, when for any $\epsilon >0$, there is a $N>0$ such that if $n\geq N$,
\[ |a_n-a|< \epsilon \]
Then we can write that:
\[ \lim_{n\rightarrow \infty }a_n=a \]
Or:
\[ a_n\rightarrow a\quad (n\rightarrow \infty ) \]
This is the most elementary expression of epsilon-delta techniques. The above example of $\lim_{n\rightarrow \infty }a_n=a$ is $2.3999\cdots  =2.4$