Prove by mathematical induction that $n^\frac{1}{n}<2-\frac{1}{n}$

Question

Could I have some guidance on how to conduct the inductive step in this proof (i.e. how to apply the assumption that P(k) holds true to prove P(k+1) )... $$n^\frac{1}{n}<2-\frac{1}{n}, \ n≥2$$

Thank you very much!

Answer 1

Hint: For $n \geq 3$, show that $ (n+1) ^ \frac{ 1}{ n+1} < n^ \frac{1}{n} < 2 - \frac{1}{n} < 2 - \frac{ 1}{ n+1}$.

So start with base case of $n=3$.

Deal with the case $n = 2 $ separately.