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 $\displaystyle \lim_{n \to \infty} x_n = l$.

Then

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

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 $n \to \infty$.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.29 \ (1) \ \text{(ii)}$