Originally Posted by

**mohammadfawaz** Hello all,

I have the following sequence: $\displaystyle a_n(x)=n(x^{\frac{1}{n}}-1)$ where $\displaystyle x>1$. The objective is to prove that it is bounded above and below. Also, we need to prove that it has no smallest element. I succeeded in proving that it is bounded below: (let $\displaystyle x=1+h$ with $\displaystyle h>0$ and continue from there). For the other questions, I think it would be enough if we can prove that the sequence is decreasing. The upper bound would be simply $\displaystyle a_1 = x-1$. I tried to prove this fact but i reached: $\displaystyle a_{n+1}-a_{n}=n(x^{\frac{1}{n+1}}-x^{\frac{1}{n}})+x^{\frac{1}{n+1}}-1$. But I'm stuck there. Help please.

Thanx