The squeeze theorem for sequences

## The squeeze theorem for sequences
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            ## Squeeze theorem
            #### Theorem
            If $\forall\\,n\in\mathbb N,\\, a_n \leqslant b_n \leqslant c_n$ and $\displaystyle\lim\_{n\to\infty} a_n = \ell = \lim\_{n\to\infty} c_n$, then also $\displaystyle\lim\_{n\to\infty} b_n = \ell$.
            #### Example 1
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$ defined by
            $\displaystyle a_n = 0$, $\displaystyle b_n = \frac1{n^3}$, $\displaystyle c_n = \frac1n$,

$\displaystyle a_n\to0$ and $\displaystyle c_n\to0$, so $\displaystyle b_n\to0$ also.

![Demonstration of the squeeze theorem example 1](gfx/squeeze-thm-1.png "Title")

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            ## Squeeze theorem
            #### Theorem
            If $\forall\\,n\in\mathbb N,\\, a_n \leqslant b_n \leqslant c_n$ and $\displaystyle\lim\_{n\to\infty} a_n = \ell = \lim\_{n\to\infty} c_n$, then also $\displaystyle\lim\_{n\to\infty} b_n = \ell$.
            #### Example 2
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$ defined by
            $\displaystyle a_n = \frac{-1}n$, $\displaystyle b_n = \frac{(-1)^n}n$, $\displaystyle c_n = \frac1n$,

again $\displaystyle a_n\to0$ and $\displaystyle c_n\to0$, so $\displaystyle b_n\to0$ also.

![Demonstration of the squeeze theorem example 2](gfx/squeeze-thm-2.png "Title")

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            ## Squeeze theorem
            #### Theorem
            If $\forall\\,n\in\mathbb N,\\, a_n \leqslant b_n \leqslant c_n$ and $\displaystyle\lim\_{n\to\infty} a_n = \ell = \lim\_{n\to\infty} c_n$, then also $\displaystyle\lim\_{n\to\infty} b_n = \ell$.
            #### Your turn
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$ defined by
            $\displaystyle a_n = ?$, $\displaystyle b_n = \frac{n^3+20}{n^7+1}$, $\displaystyle c_n = ?$,

we need $\displaystyle a_n\to0$ and $\displaystyle c_n\to0$, so that we can argue $\displaystyle b_n\to0$ also.
            
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            ## Squeeze theorem
            #### Theorem
            If $\forall\\,n\in\mathbb N,\\, a_n \leqslant b_n \leqslant c_n$ and $\displaystyle\lim\_{n\to\infty} a_n = \ell = \lim\_{n\to\infty} c_n$, then also $\displaystyle\lim\_{n\to\infty} b_n = \ell$.
            #### Your turn
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$ defined by
            $\displaystyle a_n = ?$, $\displaystyle b_n = \frac{n^3+20}{n^7+1}$, $\displaystyle c_n = ?$,

we need $\displaystyle a_n\to0$ and $\displaystyle c_n\to0$, so that we can argue $\displaystyle b_n\to0$ also.

We can use $a_n=0\to0$ and $c_n = \frac{21}n$ because $\frac{n^3+20}{n^7+1} < \frac{21 n^3}{n^7} = \frac{21}{n^4} < \frac{21}n\to0$.