# Ordered field/Convergent sequences/Rules/Fact

Rules for convergent sequences

Let ${\displaystyle {}K}$ be an ordered field and let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ be convergent sequences. Then the following statements hold.

1. The sequence ${\displaystyle {}{\left(x_{n}+y_{n}\right)}_{n\in \mathbb {N} }}$ is convergent and
${\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}+y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}+{\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}$

holds.

2. The sequence ${\displaystyle {}{\left(x_{n}\cdot y_{n}\right)}_{n\in \mathbb {N} }}$ is convergent and
${\displaystyle {}\lim _{n\rightarrow \infty }{\left(x_{n}\cdot y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}\cdot {\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,}$

holds.

3. For ${\displaystyle {}c\in K}$ we have
${\displaystyle {}\lim _{n\rightarrow \infty }cx_{n}=c{\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,.}$
4. Suppose that ${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}=x\neq 0}$ and ${\displaystyle {}x_{n}\neq 0}$ for all ${\displaystyle {}n\in \mathbb {N} }$. Then ${\displaystyle {}\left({\frac {1}{x_{n}}}\right)_{n\in \mathbb {N} }}$ is also convergent and
${\displaystyle {}\lim _{n\rightarrow \infty }{\frac {1}{x_{n}}}={\frac {1}{x}}\,}$

holds

5. Suppose that ${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}=x\neq 0}$ and that ${\displaystyle {}x_{n}\neq 0}$ for all ${\displaystyle {}n\in \mathbb {N} }$. Then ${\displaystyle {}\left({\frac {y_{n}}{x_{n}}}\right)_{n\in \mathbb {N} }}$ is also convergent and
${\displaystyle {}\lim _{n\rightarrow \infty }{\frac {y_{n}}{x_{n}}}={\frac {\lim _{n\rightarrow \infty }y_{n}}{x}}\,}$

holds.