epsilon-delta proofs

In the first post, I want to pick up epsilon-delta techniques.

In high school, it has been well said that epsilon-delta techniques have been hardly understood since students are not used to the mathematical expression. Whereas in university, as the techniques suddenly appear in limit proofs, many students are very much confused. I think that the difficulty of epsilon-delta techniques basically owes to the continuity of real number. I am going to give some hints to help you understand them.

Well, a real numerical sequence $a_n$ comes close to a value without any end. For example,

$a_0=0$
$a_1=0.9$
$a_2=0.99$
$a_3=0.999$
・・・・・

$n$ of $a_n$ means the number of 9 after the decimal point. As n increases infinitely, we can prove that this $a_n$ becomes 1. The $a_n$ when $n\rightarrow \infty $ is provided below.
\[(1) \qquad a_{\infty }=0.999\cdots \]
We multiply both sides by 10. Then,
\[(2) \qquad 10a_{\infty }=9.999\cdots  \]
subtract both sides of (1) from (2), and divide by 9,
\[ a_{\infty }=1 \]
Therefore,
\[ 1=0.999\cdots \]
 If it is possible to multiply and subtract a real number $a_{\infty }$ which has no end, we have to accept this result.

Extending this result, we come to the fact that a number is equal to the number obtained by subtracting 1 from the last digit of the finite real number and by appending $999\cdots $. For example, if $a=2.4$, it is same as $a=2.3999\cdots  $.

In conclusion, any finite real number has infinite real numerical sequences.