Combination Theorem for Limits of Functions/Real/Combined Sum Rule

Theorem
Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $f$ and $g$ be functions defined on an open subset $S \subseteq X$, except possibly at the point $c \in S$.

Let $f$ and $g$ tend to the following limits:


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

Let $\lambda, \mu \in X$ be arbitrary numbers in $X$.

Then:
 * $\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$

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

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

By the Combined Sum Rule for Sequences:
 * $\ds \lim_{n \mathop \to \infty} \paren {\lambda \map f {x_n} + \mu \map g {x_n} } = \lambda l + \mu m$

Applying Limit of Function by Convergent Sequences again, we get:
 * $\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$