In problem 1.35, we are asked to show that if sequence $p_1, p_2, p_3, ...$ converges to point$x$then $c\cdot p_1, c\cdot p_2, c\cdot p_3, ...$ convergest to $c\cdot x$.

I feel like I need to refer to definition 1.27: A sequence is a function with its domain $\mathbb{N}$and its range $\in \mathbb{R}$ and definition 1.28: $p_1, p_2, p_3, ...$ converges to x means if $S$is an open interval containing $x$ then there is a positive integer $N$such that if $n$ is a positive integer and $n\geq N$ then $p_n$$\in S$. Thus, all I really need to do is show that if $x$ is $\in\mathbb{R}$ then $c\cdot x$ is also $\in\mathbb{R}$. It makes sence but I don't see how this necessarily follows.

I can create an open interval $S=(a,b)$ containing $c\cdot x$ but how do I prove there is a $c\cdot p_n$ within that interval if $c\cdot p_n$ may not be a Real number?