Induction with nth root of n

I am trying to prove by induction that $$\sqrt[n]{n}<2-\frac{1}{n}$$ where $$n\ge2$$. It seemed simple at first, but I am stuck with $$log(2n-1)$$ in the RHS. I am in an elementary undergraduate Maths course. Please help me out.

• This is my first ever question on this platform. Help me improve. – Saswat Pati Jun 7 at 7:23
• There is a simple (non-induction) proof in this question: math.stackexchange.com/q/1322898. – Martin R Jun 7 at 7:31

You need to show , $$n\leq \left(2-\frac{1}{n}\right)^n$$ It holds for the base case $$n=2$$.

Assuming it holds for $$n=k$$,

$$k<\left(2-\frac{1}{k}\right)^k$$

we need to show the same for $$n=k+1$$ i.e.,

$$k+1<\left(2-\frac{1}{k+1}\right)^{k+1}$$

let's try to prove something stronger,

$$k+1<\left(2-\frac{1}{k}\right)^{k+1}<\left(2-\frac{1}{k+1}\right)^{k+1}$$

using the the relation for $$n=k$$,

$$\left(2-\frac{1}{k}\right)^{k+1}>\left(2-\frac{1}{k}\right)\times k=2k-1>k+1 \qquad\forall k>2$$

By Bernoulli $$\left(2-\frac{1}{n}\right)^n=\left(1+\left(1-\frac{1}{n}\right)\right)^n>1+n\left(1-\frac{1}{n}\right)=n.$$

• “I am trying to prove by induction ...” – Martin R Jun 7 at 7:40