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 $2.3999\cdots =2.4$
2012/10/27
2012/10/20
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.
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.
2012/10/17
about
Welcome to my blog, "WILL THAT BE ALL？"
This blog will contain posts on various mathematical subjects.
Nearly for 2 years, I have been posting memoranda of the probability theory on my website ( Mathematical Finance by Japanese), and have just finished it recently. Since then, I have been taking a break.
Though I have shut down a bulletin board on my website, I would like to respond to some questions, and write some articles which I gave up addressing on my website, as they are not related to the main issue.
For a change, I will also post just something in my life, impressions of books that I have finished reading recently, authors whom I have been interested in, some photos and pictures which I have taken, and so on.
Any feedback and constructive criticism would be very much appreciated.
This blog will contain posts on various mathematical subjects.
Nearly for 2 years, I have been posting memoranda of the probability theory on my website ( Mathematical Finance by Japanese), and have just finished it recently. Since then, I have been taking a break.
Though I have shut down a bulletin board on my website, I would like to respond to some questions, and write some articles which I gave up addressing on my website, as they are not related to the main issue.
For a change, I will also post just something in my life, impressions of books that I have finished reading recently, authors whom I have been interested in, some photos and pictures which I have taken, and so on.
Any feedback and constructive criticism would be very much appreciated.