Real sequence/Positive/Divergent to infinity and inverse null sequence/Exercise

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence with ${}x_{n}>0$ for all ${}n\in \mathbb {N}$.

Prove that the sequence diverges to ${}+\infty$ if and only if the sequence ${}{\left({\frac {1}{x_{n}}}\right)}_{n\in \mathbb {N} }$ converges to ${}0$.

Create a solution