# Squeeze Theorem/Sequences/Real Numbers

## Theorem

Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$.

Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit:

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

Suppose that:

- $\forall n \in \N: y_n \le x_n \le z_n$

Then:

- $x_n \to l$ as $n \to \infty$

that is:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = l$

Thus, if $\sequence {x_n}$ is always between two other sequences that both converge to the same limit, $\sequence {x_n} $ is said to be **sandwiched** or **squeezed** between those two sequences and itself must therefore converge to that same limit.

### Corollary

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

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

Let:

- $\forall n \in \N: \size {x_n - l} \le y_n$

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

## Proof

From Negative of Absolute Value: Corollary 1:

- $\size {x - l} < \epsilon \iff l - \epsilon < x < l + \epsilon$

Let $\epsilon > 0$.

We need to prove that:

- $\exists N: \forall n > N: \size {x_n - l} < \epsilon$

As $\displaystyle \lim_{n \mathop \to \infty} y_n = l$ we know that:

- $\exists N_1: \forall n > N_1: \size {y_n - l} < \epsilon$

As $\displaystyle \lim_{n \mathop \to \infty} z_n = l$ we know that:

- $\exists N_2: \forall n > N_2: \size {z_n - l} < \epsilon$

Let $N = \max \set {N_1, N_2}$.

Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.

So:

- $\forall n > N: l - \epsilon < y_n < l + \epsilon$
- $\forall n > N: l - \epsilon < z_n < l + \epsilon$

But:

- $\forall n \in \N: y_n \le x_n \le z_n$

So:

- $\forall n > N: l - \epsilon < y_n \le x_n \le z_n < l + \epsilon$

and so:

- $\forall n > N: l - \epsilon < x_n < l + \epsilon$

So:

- $\forall n > N: \size {x_n - l} < \epsilon$

Hence the result.

$\blacksquare$

## Also known as

This result is also known, in the UK in particular, as the **sandwich theorem** or the **sandwich rule**.

In that culture, the word **sandwich** traditionally means specifically enclosing food between two slices of bread, as opposed to the looser usage of the **open sandwich**, where the there is only one such slice.

Hence, in idiomatic British English, one can refer to the (often uncomfortable) situation of being between two entities as being **sandwiched** between them.

As the idiom is not universal globally, the term **squeeze theorem** is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$, for greatest comprehension.

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.10$: Theorem (the sandwich theorem)