# Real sequence/Convergence/Unit fractions/Exercise

Let be a real sequence. Prove that the sequence converges to if and only if for all a natural number exists, such that for all the estimation holds.

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