Proving a term converges pointwise by the definition

Question

How would I show
$$f_n(x) = n^\alpha x^n (1-x)\qquad \text{for} \: x \in [0,1]$$ converges pointwise to $0$ by the definition?
I showed that it converges for $x=0,1$ trivially, but I'm not sure how I would use the definition for the interior of the domain.

Answer 1

Note that $|f_n(x)| \leq n^{\alpha}|x|^n$ for $x \in (0,1)$, then you need to recall that $\lim_{n \to \infty}{\frac{n^{\alpha}}{x^n}} \to 0$ whenever $x > 1$.