Mathematical Analysis study notes[1]

real numbers and limit

1. a number $x$ can be call real number due to that it must be represented with $x=\frac{a}{b}$when the $a,b$ are integer numbers and $b\ne 0$.
2. a real number is closed under all arithmetic operations such as addition,subtraction,multiplation and division.moreover,the divisor can not be zero.
3. the convergent sequence of numbers $\{x_n\}$ is $x$ when $n\to \infty$,$x_n\to x$,for example,the limitation of sequence $\{1+\frac{1}{n}\}$ is $1$.
4. there is a formal explanation that let $\{x_n\}$ is a sequence of numbers and $x$ is a real number.if it is true that for giving arbitrary $ϵ>0$,$N\in {\mathbb{N}}^{+}$ exists and $|x_n-x|<ϵ$ when $n>N$,the sequence $\{x_n\}$ is called convergent sequence of numbers,otherwise it is known as divergent sequence.by the way,the limitation of convergent sequence of numbers is exclusive.
5. the $ϵ$ neighborhood of x is open interval$(a-ϵ,a+ϵ)$
6. if existing a real number x result in $x_n\le x$ for $x_n$,the x can be called upper bound.defining lower bound as similar as this.if a sequence have both upper bound and lower bound ,so it can be called bounded sequence .

references

1. 《数学分析教程》