In the book of MATHEMATICAL METHODS FOR PHYSICISTS by Arfken et al., (Example 1.2.1 Non uniform convergence):
The series of functions $S(x)=\sum_{n=0}^{\infty} \left(1-x\right)x^n$ is shown to not be uniformly convergent by providing the following explanation: $$|1-(1-x^N)|<ε$$
"No matter what the values of N and a sufficiently small ε may be, there will be an x value (close to 1) where this criterion is violated. The underlying problem is that x = 1 is the convergence limit of the geometric series, and it is not possible to have a convergence rate that is bounded independently of x in a range that includes x = 1."
Can anyone please explain why is it possible to have such an $x$ value which will always invalidate the inequality?
HINT:
What happens if $x=1-1/N$?
Alternatively, take $\epsilon=\frac12$ and $x=\sqrt[N]{\frac{1}{2}}$.