Let ( x n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }} be a real sequence. Prove that the sequence converges to x {\displaystyle {}x} if and only if for all k ∈ N + {\displaystyle {}k\in \mathbb {N} _{+}} a natural number n 0 ∈ N {\displaystyle {}n_{0}\in \mathbb {N} } exists, such that for all n ≥ n 0 {\displaystyle {}n\geq n_{0}} the estimation | x n − x | ≤ 1 k {\displaystyle {}\vert {x_{n}-x}\vert \leq {\frac {1}{k}}} holds.
