Question

Use the definition of limit to prove that the sequence { 1 √ n+1} converges to 0.

Answer 1

$Let \space \epsilon >0$

We want $\mid\frac{1}{n^{0.5}}-0\mid<\epsilon\iff\frac{1}{n^{0.5}}<\epsilon\iff{n^{0.5}}>\frac{1}{\epsilon}$

So that $n>\frac{1}{{\epsilon}^{0.5}}$

Taking $N = \frac{1}{{\epsilon}^{0.5}}$ we have n>N $\implies n>\frac{1}{{\epsilon}^{0.5}}$ and for which we have

$\mid\frac{1}{n^{0.5}}-0\mid<\epsilon$

Meaning $\lim\limits_{x\to \infin}\frac{1}{n^{0.5}}=0$ by the definition of limit