Uniform convergence of $f_n(x)=nx^n(1-x^n)$ on $[0,1]$

Question

I want to check the uniform convergence of the sequence of function $f_n(x)=nx^n(1-x^n)$ on $[0,1]$.

My approach: We can see $f_n$ converges pointwise to $f(x)=0$

Now If we consider the sequence $x_k=(1/k)^\frac{1}{k}$ in $[0,1]$ then $f_n(x_n)=n\frac{1}{n}(1-\frac{1}{n})$ ,therefore $M_n=\sup|f_n(x)|>1$, which is nonzero, so it is not uniformly convergent.

Is this correct? Or is there any easy way to handle this kind of problem?