Complex Sequence is Null iff Modulus of Sequence is Null
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Theorem
Let $\sequence {z_n}_{n \mathop \in \N}$ be a complex sequence.
Then:
- $z_n \to 0$
- $\cmod {z_n} \to 0$
That is:
- $\sequence {z_n}_{n \mathop \in \N}$ is a null sequence if and only if $\sequence {\cmod {z_n} }_{n \mathop \in \N}$ is a null sequence.
Proof 1
By the definition of convergent sequence, we have:
- $z_n \to 0$ if and only if for each $\epsilon > 0$ there exists $N \in \N$ such that $\cmod {z_n} < \epsilon$ for each $n \ge N$.
Similarly:
- $\cmod {z_n} \to 0$ if and only if for each $\epsilon > 0$ there exists $N \in \N$ such that $\cmod {\paren {\cmod {z_n} } } < \epsilon$ for each $n \ge N$.
This article, or a section of it, needs explaining. In particular: What does $\cmod {\paren {\cmod {z_n} } }$ mean? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
From Complex Modulus of Real Number equals Absolute Value, we have:
- $\cmod {\paren {\cmod {z_n} } } = \cmod {z_n}$
This article, or a section of it, needs explaining. In particular: How does the above equation follow from Complex Modulus of Real Number equals Absolute Value? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
So:
- $\cmod {z_n} \to 0$ if and only if for each $\epsilon > 0$ there exists $N \in \N$ such that $\cmod {z_n} < \epsilon$ for each $n \ge N$.
We can therefore see that:
- $z_n \to 0$
- $\cmod {z_n} \to 0$
$\blacksquare$
Proof 2
Observe:
\(\text {(1)}: \quad\) | \(\ds \cmod {z_n - 0}\) | \(=\) | \(\ds \cmod {z_n}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {z_n} - 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\cmod {z_n} - 0}\) |
Therefore:
\(\ds \lim_{n \to \infty} z_n\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall \epsilon \in \R_{>0}: \exists N \in \R: \, \) | \(\ds n > N\) | \(\implies\) | \(\ds \cmod {z_n - 0} < \epsilon\) | Definition of Convergent Complex Sequence | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \forall \epsilon \in \R_{>0}: \exists N \in \R: \, \) | \(\ds n > N\) | \(\implies\) | \(\ds \cmod {\cmod {z_n} - 0} < \epsilon\) | from $\paren 1$ | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \lim _{n \to \infty} \cmod {z_n}\) | \(=\) | \(\ds 0\) | Definition of Convergent Complex Sequence |
$\blacksquare$