Squeeze Theorem for Sequences

Theorem

There are two versions of this result:

• one for sequences in the set of complex numbers $\C$
• one for sequences in the set of real numbers $\R$ (which is stronger).

Otherwise known (particularly in the UK) as the sandwich theorem.

Sequences of Real Numbers

Let $\left \langle {x_n} \right \rangle, \left \langle {y_n} \right \rangle$ and $\left \langle {z_n} \right \rangle$ be sequences in $\R$.

Let $\left \langle {y_n} \right \rangle$ and $\left \langle {z_n} \right \rangle$ be convergent to the following limit:

$\displaystyle \lim_{n \to \infty} y_n = l, \lim_{n \to \infty} z_n = l$

Suppose that $\forall n \in \N: \left \langle {y_n} \right \rangle \le \left \langle {x_n} \right \rangle \le \left \langle {z_n} \right \rangle$.

Then $x_n \to l$ as $n \to \infty$, that is, $\displaystyle \lim_{n \to \infty} x_n = l$.

That is, if $\left \langle {x_n} \right \rangle$ is always between two other sequences that both converge to the same limit, $\left \langle {x_n} \right \rangle$ is said to be sandwiched or squeezed between those two sequences and itself must therefore converge to that same limit.

Corollary

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

$y_n \to 0$ as $n \to \infty$

Let:

$\forall n \in \N: \left|{x_n - l}\right| \le y_n$

Then $x_n \to l$ as $n \to \infty$.

Sequences of Complex Numbers

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.