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