
$\{p_n\}$ converges to $p$:$\displaystyle\lim_{n \to \infty}p_n=p$: $p_n\rightarrow p$:
\\
$\forall ε>0\exists n_0\in \mathbb{N}\forall n\in \mathbb{N}(n\geq n_0\rightarrow d(p_n,p)<ε)$


Limit of sequences

$\{s_n\}:$が$s$に収束:$\displaystyle\lim_{n \to \infty}s_n=s $または $ s_n\rightarrow s$
\\
$\forallε>0 ,\exists N \in \mathbb{N} \forall n\in \mathbb{N} ,(n \ge N \rightarrow |s_n-s|<ε)$



Limit of real sequence


$
\lim_{n \to \infty} q^n=
  \begin{cases}
0 \,\,\,\,\,\,(0<q<1) 
\\
1 \,\,\,\,\,\,(q=1)
\\
∞ \,\,\,\,(q>1)
\end{cases}
$

Corollary

A real number sequence that convege is called a convergent real number sequence.

Convergent real number sequence

$\begin{aligned}
&\{s_n\}\text{ is bounded:}\exists B >0 ,\forall n\in \mathbb{N}(|s_n|<B)
\\
&\{s_n\}\text{ is bounded to the above:}\exists M >0 ,\forall n\in \mathbb{N}(s_n<M)
\\
&\{s_n\}\text{ is bounded to the below:}\exists L \ >0 ,\forall n\in \mathbb{N}(s_n>L)
\end{aligned}$$




Bounded sequence


$\{s_n\}$ diverges :$\{s_n\}$ is not a convergent sequence

Diverge

$\{s_n\}$ diverge to $∞:\lim_{n \to \infty} s_n=∞$:
\\$\forall B>0,\exists N \in \mathbb{N},\forall n \in \mathbb{N}, (n≥N \rightarrow s_n>B)$
$


Convergence of real sequence

$Limit of sequences$

$Limit of real sequence$

$Corollary$

$Convergent real number sequence$

$Bounded sequence$

$Corollary$

$Diverge$

$Diverge to infinity$