Proving that a set is a set of generators

Homework Statement

I want to show that the set
$$<1,x,x^2,\cdots ,x^n>$$
forms a basis of the space
$$P_{n}$$
where
$$P_{n}$$ contains all polynomial functions up to fixed degree n.

The Attempt at a Solution

I have already shown that the set
$$<1,x,x^2,\cdots ,x^n>$$
is linearly independent and now I want to show that this set is a set of generators for $$P_{n}.$$

Take any
$$f\in P_{n}.$$ Let
$$<\alpha_{0},...,\alpha_{n}>$$
represent the coefficients of $$f.$$ Then since
$$\alpha_{0}\cdot 1=\alpha_{0},...,\alpha_{n}\cdot x^{n}=\alpha_{n}x^{n}$$
$$\alpha_{0}+\cdots+\alpha_{n}x^{n}=f.$$

Is that correct or am I missing something? Thanks!

tiny-tim