Convergence of Modulus of Convergent Complex Sequence
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Theorem
Let $\sequence {z_n}$ be a sequence in $\C$.
Let $\sequence {z_n}$ converge to a value $c \in \C$.
Let $\cmod z$ denote the modulus of a complex number $z$.
Then:
- $\sequence {\cmod {z_n} }$ converges to a value $\cmod c$.
Proof
By definition of convergent complex sequence:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$
From the Reverse Triangle Inequality:
- $\size {\cmod x - \cmod y} \le \cmod {x - y}$
and the result follows.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Corollary.