# Real numbers/Convergent sequences/Rules/3/Fact/Proof/Exercise

Let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ be a real convergent sequence and ${\displaystyle {}c\in \mathbb {R} }$. Show that the sequence ${\displaystyle {}{\left(c\cdot x_{n}\right)}_{n\in \mathbb {N} }}$ is also convergent and
${\displaystyle {}\lim _{n\rightarrow \infty }{\left(c\cdot x_{n}\right)}=c\cdot {\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,}$