Sandwich principle

**simplysparklers** · May 12th 2008, 10:11 AM

Hi guys!
Ok, last problem today! I've already read any other posts on the sandwich principle, but I've never seen it before today, and my chapter doesn't explain it, just gives a question on it!

If anyone could show me how this is done I would really appreciate it!

Let $(a_{n}),(x_{n})$ and $(b_{n})$ be sequences with limits $\alpha,\xi,\beta$ respectively, and suppose that, for all $n\geq{1}$ ,
$a_{n}\leq{x_{n}}\leq{b_{n}}$

Show that $\alpha\leq\xi\leq\beta$ [This is sometimes called the sandwich principle. It can be useful if $(a_{n})$ and $(b_{n})$ are "known" sequences and $(x_{n})$ is unknown. It is especially useful when $\alpha=\beta$ , for in this case we conclude that $\xi=\alpha=\beta$ .]

Nice explaination, if I did actually know what $(a_{n})$ and $(b_{n})$ were!In this case I don't know what to do, can anyone help please?

**Plato** · May 12th 2008, 11:31 AM

Let’s do it for one half, you can expand.
Suppose $\xi < \alpha$ then let $\varepsilon = \frac{{\alpha - \xi }}{2} > 0$ .
Use the sequence convergence definition twice to get $\left( {\exists N} \right)\left[ {k \geqslant N \Rightarrow \quad \left| {a_k - \alpha } \right| < \varepsilon \,\& \,\left| {x_k - \xi } \right| < \varepsilon } \right]$ .
Removing the absolute values and using the given we get: $\frac{{\xi + \alpha }}<br /> {2} < a_N \leqslant x_N < \frac{{\xi + \alpha }}{2}$ .

You can see the contradiction.
Now do it again for the other half.

**arbolis** · May 12th 2008, 12:11 PM

As you said, suppose that ${a_n}\leq{b_n}$ for all $n\geq1$ .
I give you an example. Usually the exercise will make you calcul the limit of the sequences ${a_n}$ and $b_n$ . Suppose now that you found that $a_n$ tends to 10 (from below) when $n$ tends to positive infinite and $b_n$ tends to 10 (from bottom) when $n$ tends to positive infinite. If you can see that $a_n\leq x_n$ and $x_n\leq b_n$ for a given sequence $x_n$ , then by the sandwich's theorem (or principle), $x_n$ tends to 10, which is very intuitive!

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