Squeeze Theorem/Sequences/Real Numbers

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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