Modulus of Limit

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

Let $\sequence {x_n}$ be a sequence in $X$.

Let $\sequence {x_n}$ be convergent to the limit $l$.

That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$.

Then
 * $\displaystyle \lim_{n \mathop \to \infty} \cmod {x_n} = \cmod l$

where $\cmod {x_n}$ is the modulus of $x_n$.

Proof
By the Triangle Inequality, we have:
 * $\cmod {\cmod {x_n} - \cmod l} \le \cmod {x_n - l}$

Hence by the Squeeze Theorem and Convergent Sequence Minus Limit, $\cmod {x_n} \to \cmod l$ as $n \to \infty$.