Combination Theorem for Limits of Functions/Real/Product 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:


 * $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$
 * $\displaystyle \lim_{x \mathop \to c} g \left({x}\right) = m$

Then:
 * $\displaystyle \lim_{x \mathop \to c} \ \left({f \left({x}\right) g \left({x}\right)}\right) = l m$

Proof
Let $\left \langle {x_n} \right \rangle$ be any sequence of points of $S$ such that:
 * $\forall n \in \N^*: x_n \ne c$
 * $\displaystyle \lim_{n \mathop \to \infty} x_n = c$

By Limit of Function by Convergent Sequences:
 * $\displaystyle \lim_{n \mathop \to \infty} \ f \left({x_n}\right) = l$
 * $\displaystyle \lim_{n \mathop \to \infty} \ g \left({x_n}\right) = m$

By the Product Rule for Sequences:
 * $\displaystyle \lim_{n \mathop \to \infty} \ \left({f \left({x_n}\right) g \left({x_n}\right)}\right) = l m$

Applying Limit of Function by Convergent Sequences again, we get:
 * $\displaystyle \lim_{x \mathop \to c} \ \left({f \left({x}\right) g \left({x}\right)}\right) = l m$