$f_n=\frac{1}{x^{\frac{1}{n}}}$ uniformly converges to 1

Question

This is the problem that I am trying to prove. Show that $f_n(x)=\frac{1}{x^{\frac{1}{n}}}$ where $x\in (0,\frac{1}{2})$ is uniformly convergent to $f(x)=1$. It will be great if someone can tell me how to prove this.

Answer 1

Unfortunately, I think this is wrong to say the convergence on $x\in (0,\frac{1}{2})$ is uniform. (You should carefully read again what Dan commented) , since, for any fixed $n$, $$\sup_{x\in(0,\frac{1}{2})}|f_n(x)-1|=\sup_{x\in(0,\frac{1}{2})} (\frac{1}{\sqrt[n]{x}}-1)=+\infty$$, which does not converge to $0$ as $n\to\infty$.