Prove by definition $\lim_{n\to\infty} \sqrt{\frac{n}{n+1}}=1$

Question

Prove by definition $\lim_{n\to\infty} \sqrt{\frac{n}{n+1}}=1$.

I got to $\left|\sqrt{\frac{n}{n+1}}-1\right|< ε$, but I don't know how to proceed.

Answer 1

Let me explain Kenny Lau's answer a bit. The first step, he does

$$\left|\sqrt{\frac{n}{n+1}}-1\right|=\left|\frac{\sqrt{n}}{\sqrt{n+1}}-1\right|=\left|\frac{\sqrt{n}}{\sqrt{n+1}}-\frac{\sqrt{n+1}}{\sqrt{n+1}}\right|=\left|\frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n+1}}\right|$$

Now he proceeds to multiply both sides of the fraction with $\sqrt{n}+\sqrt{n+1}$; note that $(a-b)(a+b)=a^2-b^2$ so that the numerator becomes

$$(\sqrt{n}-\sqrt{n+1})(\sqrt{n}+\sqrt{n+1})=\sqrt{n}^2-\sqrt{n+1}^2=n-(n+1)=-1$$

so that we get

$$\left|\frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n+1}}\right|=\left|\frac{-1}{{\sqrt{n+1}(\sqrt{n}+\sqrt{n+1})}}\right|$$

He discards the $-$-sign because we're taking the absolute value anyways. Now he makes the point that $\sqrt{n}<\sqrt{n+1}$ so that

$$\sqrt{n+1}(\sqrt{n}+\sqrt{n+1})>\sqrt{n}(\sqrt{n}+\sqrt{n})=2n$$

hence,

$$\frac{1}{\sqrt{n+1}(\sqrt{n}+\sqrt{n+1})}<\frac{1}{\sqrt{n}(\sqrt{n}+\sqrt{n})}=\frac{1}{2n}$$

so in the end

$$\left|\sqrt{\frac{n}{n+1}}-1\right|<\left|\frac{1}{2n}\right|$$

the point he makes with this, is that you can always pick $n$ big enough so that it gets smaller than a certain $\epsilon$, because you can just make $\frac1{2n}$ smaller and then

$$\left|\sqrt{\frac{n}{n+1}}-1\right|$$

must be smaller too.

Answer 2

$$\begin{array}{rcl} \left| \sqrt{\dfrac n {n+1}} - 1 \right| &=& \left| \dfrac {\sqrt n - \sqrt{n+1}} {\sqrt{n+1}} \right| \\ &=& \left| \dfrac 1 {\sqrt{n+1} (\sqrt n + \sqrt {n+1})} \right| \\ &\le& \left| \dfrac 1 {\sqrt{n} (\sqrt n + \sqrt {n})} \right| \\ &=& \left| \dfrac 1 {2n} \right| \\ \end{array}$$