Combination Theorem for Limits of Functions/Real/Multiple Rule

Theorem
Let $\R$ denote the real numbers.

Let $f$ be a real function defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.

Let $f$ tend to the following limit:


 * $\ds \lim_{x \mathop \to c} \map f x = l$

Let $\lambda \in \R$ be an arbitrary real number.

Then:
 * $\ds \lim_{x \mathop \to c} \paren {\lambda f \paren x} = \lambda l$

Proof
Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:
 * $\forall n \in \N_{>0}: x_n \ne c$
 * $\ds \lim_{n \mathop \to \infty} x_n = c$

By Limit of Real Function by Convergent Sequences:
 * $\ds \lim_{n \mathop \to \infty} \map f {x_n} = l$

By the Multiple Rule for Sequences:
 * $\ds \lim_{n \mathop \to \infty} \paren {\lambda \map f {x_n} } = \lambda l$

Applying Limit of Real Function by Convergent Sequences again:
 * $\ds \lim_{x \mathop \to c} \paren {\lambda \map f x} = \lambda l$