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

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

${}\lim _{n\rightarrow \infty }{\left(c\cdot x_{n}\right)}=c\cdot {\left(\lim _{n\rightarrow \infty }x_{n}\right)}\,$

holds.

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