Squeeze Theorem/Sequences/Complex Numbers

Theorem
Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$ which is null, that is:
 * $a_n \to 0$ as $n \to \infty$

Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.

Suppose $\left \langle {a_n} \right \rangle$ dominates $\left \langle {z_n} \right \rangle$.

That is, suppose that:
 * $\forall n \in \N: \left|{z_n}\right| \le a_n$

Then $\left \langle {z_n} \right \rangle$ is a null sequence.

Proof
In order to show that $\left \langle {z_n} \right \rangle$ is a null sequence, we want to show that:
 * $\forall \epsilon > 0: \exists N: \forall n > N: \left|{z_n}\right| < \epsilon$

But since $\left \langle {a_n} \right \rangle$ is a null sequence:
 * $\exists N: \forall n > N: a_n < \epsilon$

So, using this value of $n$, we have: $\left \vert {z_n} \right \vert \le a_n < \epsilon$

Thus $\left\langle{ z_n }\right\rangle$ is a null sequence.

Also known as
This result is also known, in the UK in particular, as the sandwich theorem.