Question

Use the definition of limit to prove that both of the sequences { 1 √ n √ } and {(−1)n n } converges to 0.

Answer 1

Let $a_{n}=\frac{1}{\sqrt{n}}$ . Let $\epsilon>0$ is given. Then there exist some N>0 satisfying $N>\frac{1}{\epsilon^{2}}$.( Archimedean property of real numbers)

Therefore $\forall$ $n\geq N$,

$|a_{n}-0|=|\frac{1}{\sqrt{n}}|=\frac{1}{\sqrt{n}}\leq \frac{1}{\sqrt{N}}<\epsilon$ .

Hence from the definition of limits $a_{n} \rightarrow 0.$

Let $b_{n}= \frac{(-1)^{n}}{n}.$ Then $\forall n\geq N$,

$|b_{n}-0|=|\frac{(-1)^{n}}{n}|=\frac{1}{n}\leq \frac{1}{N}<\epsilon$ .

Hence from the definition of limits $b_{n}\rightarrow 0.$