show $P_n(F)$ is generated by $\{1,x,...,x^n\}$

Question

i'm unsure about the quality of this proof. i'm trying to generalize the approach that i use with a given set of polynomials to show that it generates another polynomial or all polynomials of a certain degree. my attempt begins on the next line.

let $a_i \in F, i=0,...,n$

let $f$ be an nth degree polynomial, so $f \in P_n(F)$ and $f=a_0+a_1x+a_2x^2+...+a_nx_n$

let $b_j \in F, j=0,...,n$, and let $b_j$ be the coefficients of the linear combination of all elements in $\{1,x,...,x^n\}$, i.e.,

$b_0+b_1x+b_2x^2+...+b_nx^n$

combining the generic polynomial with the generating one as an identity gives:

$a_0+a_1x+a_2x^2+...+a_nx_n=b_0+b_1x+b_2x^2+...+b_nx^n$

the above result implies $a_i=b_j, (i,j=0,...,n)$

therefore $P_n(F)$ is generated by $\{1,x,...,x^n\}$.