Real numbers/Convergent sequences/Rules/Fact

Rules for convergent sequences

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

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

   ${}\lim _{n\rightarrow \infty }{\left(x_{n}+y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}+{\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,$

   holds.

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

   ${}\lim _{n\rightarrow \infty }{\left(x_{n}\cdot y_{n}\right)}={\left(\lim _{n\rightarrow \infty }x_{n}\right)}\cdot {\left(\lim _{n\rightarrow \infty }y_{n}\right)}\,$

   holds.

3. For ${}c\in \mathbb {R}$, we have

   ${}\lim _{n\rightarrow \infty }cx_{n}=c{\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,.$

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

   ${}\lim _{n\rightarrow \infty }{\frac {1}{x_{n}}}={\frac {1}{x}}\,$

   holds.

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

   ${}\lim _{n\rightarrow \infty }{\frac {y_{n}}{x_{n}}}={\frac {\lim _{n\rightarrow \infty }y_{n}}{x}}\,$

   holds.

Proof, Write another proof