Uniform convergence on [0,a]

Question

On the interval $[0,1)$ there is a function sequence $(f_n)$ given by

$f_n=\frac {1} {1-x^n}$ , $x\in [0,1)$.

Now i want to show that for every $a\in (0,1)$, $f_n$ converge uniformly on $[0,a]$. Now I know that it does not converge on the interval [0,1) because the derivative would not allow us to go to 0 when $n$ tends to infinity. But i thought that uniform convergence depends on the lowest value within the interval in this case 0. What am i doing wrong?

Answer 1

Just note that $$ |\frac{1}{1-x^{n}} - 1| \leq |\frac{1}{1 - a^{n}} - 1| \to 0 $$ as $n \to \infty$ for all $x \in [0,a]$.