Real numbers/Convergent sequences/Rules/5/Fact/Proof/Exercise

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

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

holds.

Create a solution