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 $x_n\ne 0$ for all $n\in \mathbb{N}$ and $\underset{n\to \mathrm{\infty }}{lim}{x}_n=x\ne 0$. Show that ${\left(\frac{1}{x_n}\right)}_{n\in \mathbb{N}}$is also convergent and

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{x_n}=\frac{1}{x}$

holds.

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