Squeeze Theorem for Sequences/Complex Numbers

Theorem

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$ which is null, that is:

$y_n \to 0$ 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: $\left \langle {z_n} \right \rangle \le a_n < \epsilon$

Hence the result.

$\blacksquare$