Real numbers/Convergent sequences/Rules/4/Fact/Proof/Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real convergent sequence such that ${}x_{n}\neq 0$ for all ${}n\in \mathbb {N}$ and ${}\lim _{n\rightarrow \infty }x_{n}=x\neq 0$. Show that ${}{\left({\frac {1}{x_{n}}}\right)}_{n\in \mathbb {N} }$is also convergent and

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

holds.

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