Question

Let be any positive integer. Use sandwich theorem for sequences to prove that the sequence { 1 n} converges to 0.

Answer 1

We know that $\{a_n\}$ converges to L, or, $\lim_{n \rightarrow \infty}=L$ if $\forall$ $\epsilon$ $>0$ $\exist$ N$\in$ $Z^+$ such that $\forall$ $n>N$,

$|a_n-L|<\epsilon$ .

Consider $|\dfrac{1}{ n}-0|=|\dfrac{1}{ n}|=\dfrac{1}{ n}<\epsilon$

$\Rightarrow 1<\epsilon\cdot n \Rightarrow \dfrac{1}{\epsilon}< n \Rightarrow n>\dfrac{1}{\epsilon}$

Proof: Let $\epsilon$ $>0$ . Choose N$>\dfrac{1}{\epsilon}$ . Then, $\forall$ $n>N$, $|\dfrac{1}{ n}-0|=|\dfrac{1}{ n}|$

Also, $n>N>\dfrac{1}{\epsilon} \Rightarrow \epsilon>\dfrac1 n \Rightarrow {\dfrac1n}<\epsilon \Rightarrow {\dfrac1{ n}}<\epsilon$

Thus, $|\dfrac{1}{ n}-0|=\dfrac{1}{ n}<\epsilon$

Hence, by above statement, $\{\dfrac{1}{n}\}$ converges to 0.