Modulus of Limit

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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 $x_n$.


Normed Division Ring

Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring.

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

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


Then

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


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 $n \to \infty$.

$\blacksquare$


Sources