Let ( x n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }} be a real sequence with x n > 0 {\displaystyle {}x_{n}>0} for all n ∈ N {\displaystyle {}n\in \mathbb {N} } . Prove that the sequence diverges to + ∞ {\displaystyle {}+\infty } if and only if the sequence ( 1 x n ) n ∈ N {\displaystyle {}{\left({\frac {1}{x_{n}}}\right)}_{n\in \mathbb {N} }} converges to 0 {\displaystyle {}0} .
