Sandwich Theorem

analysis

Definition

Sandwich Theorem

Let $(a_n{)}_{n\ge 0}$ and $(b_n{)}_{n\ge 0}$ be convergent sequences with identical limits ${\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }{b}_n$. Let $(c_n{)}_{n\ge 0}$ a sequence with:

$$a_n\le c_n\le b_n$$

for almost all $n\in \mathbb{N}$. Then the convergence of $(c_n{)}_{n\ge 0}$ and it holds that:

$$\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{c}_n=\underset{n\to \infty }{\lim }{b}_n$$

Proof

Sandwich Theorem

Let $(a_n{)}_{n\ge 0}$ and $(b_n{)}_{n\ge 0}$ be convergent sequences with identical limits $\ell ={\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }{b}_n$. Let $(c_n{)}_{n\ge 0}$ a sequence with:

$$a_n\le c_n\le b_n$$

For $\epsilon >0$ it holds that $a_n\in U_{\epsilon }(\ell )$, if $n>N_1$, and $b_n\in U_{\epsilon }(\ell )$, if $n>N_2$. Then it follows that $c_n\in U_{\epsilon }(\ell )$, if $n>\max (N_1,N_2)$.

$\square$