Modulus of Limit

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

Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $X$.

Let $$\left \langle {x_n} \right \rangle$$ be convergent to the limit $$l$$.

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

Then
 * $$\lim_{n \to \infty} \left|{x_n}\right| = \left|{l}\right|$$

where $$\left|{x_n}\right|$$ is the modulus of $$x_n$$.

Proof
By the Triangle Inequality, we have $$\left|{\left|{x_n}\right| - \left|{l}\right|}\right| \le \left|{x_n - l}\right|$$.

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