I have a little struggle to show pointwise and uniform convergence of $ g_n (x) = \sqrt[n] {x} $ in the intervals of $(0, \infty), (0,1), [1, \infty) $
For $lim_{n \rightarrow \inf} g_n(x) = \begin{cases} 0 , for x=0 \\ 1,for 0 < x \leq 1 \end{cases} $
So in all those intervals the function does converge pointwisely.
Now my attempt for $(0,1)$ for uniform convergence: $| g_n(x) - g(x) |= | \sqrt[n] {x} - 1| \leq \epsilon$
In the next step I converted the inequation to n : $ \frac{ln(x)} {ln( \epsilon +1)} \geq n$
What does it tell me? And how do I proof uniform convergence in $(0, \infty), [1, \infty) $? Any help really appreciated.