Show uniform convergence of sequence of functions $f_n(x)=x^n$ on $[0,0.9]$

Question

Show uniform convergence of sequence of functions $f_n(x)=x^n$ on [0, 0.9]. There is a similar question for the interval [0, 1) (https://math.stackexchange.com/questions/2191621/is-the-sequence-of-functions-f-nx-xn-uniformly-convergent-on-0-1). In this case it is not uniformly convergent. Now I need to show that it is uniformly convergent on [0, 0.9]. But I don't see the difference. Let's say if we have $x = 0.9^{1/n}$ then $x^n = 0.9$ and if we pick $\epsilon = 1/2$ then $x^n > \epsilon$. So where is my error?