# uniform convergence of $f_n(x)=x^{n-x/n}$ in $(0,1)$

Given this succession of functions I have proved the pointwise convergence to the null function.

For the uniform convergence I calculate $$\lim_{x\rightarrow 0^+} f_n(x)=0$$ and $$\lim_{x\rightarrow 1^-} f_n(x)=1$$ so can I say sup$$f_n(x)\ge1$$ in $$(0,1)$$ and there isn't uniform convergence?

So exists a sequence $$x_m \to 1$$ such that $$f_n(x_m) \to 1$$
Thus $$\sup_{x \in (0,1)}|f_n(x)|\geq \sup_{x \in \{x_1,x_2,...x_m,..\}}|f_n(x)| \geq 1$$