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:

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

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 Real Sequences:

$\ds \lim_{n \mathop \to \infty} \lambda \map f {x_n} = \lambda l$

Applying Limit of Real Function by Convergent Sequences again:

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

$\blacksquare$