# Real numbers/Convergent sequence/Average sequence/Exercise

Let be a convergent real sequence with limit . Show that the sequence defined by

also converges to .

Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a convergent real sequence with limit ${}x$. Show that the sequence defined by

- ${}y_{n}:={\frac {x_{0}+x_{1}+\cdots +x_{n}}{n+1}}\,$

also converges to ${}x$.