Using Bernoulli's Inequality to prove $n^{\frac{1}{n}} < 2-\frac{1}{n}$

Question

I was trying to prove that $$n^{\frac{1}{n}}<2-\frac{1}{n}$$ for all natural numbers $n \ge 2$.

The base case of n = 2 was trivial.

Looking at the $n+1$ case, I wrote that $$(n+1)^{\frac{1}{1+n}} \ge 1 + \frac{n}{n+1}$$ for some $n>2$, but I wasn't sure how to proceed from here

Answer 1

Write $$\Bigl(2-\frac1n\Bigr)^n=\biggl(1+\Bigl(1-\frac1n\Bigr)\biggr)^n >1+n\Bigl(1-\frac1n\Bigr)=1+n-1.$$

Answer 2

Better write $$(2-{1\over n})^n>n$$

We have $$(2-{1\over n})^n = 2^n(1-{1\over 2n})^n \geq 2^n(1-n\cdot {1\over 2n})=2^{n-1}$$

Now try with induction (for $n>2$) $2^{n-1}>n$...