Combination Theorem for Limits of Mappings/Metric Space/Sum Rule

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $\R$ denote the real numbers.

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


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

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

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

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 Sum Rule for Sequences in Metric Space:
 * $\ds \lim_{n \mathop \to \infty} \paren {\map f {x_n} + \map g {x_n} } = l + m$

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