Prove that every set formed by polynomials of different degrees is linearly independent How to prove that every set formed by polynomials of different degrees is linearly independent.
My problem is in how to make a general set of polynomials of different degrees. 

I would order them in increasing degree.  Then, if $\langle S \rangle$ denotes the subspace generated by a subset $S$, I would use the following fact:
FACT:  If $u_1, \ldots , u_r$ are linearly independent and $u_{r+1}$ does not belong to $\langle u_1, \ldots, u_r\rangle$, then $u_1, \ldots , u_{r+1}$ are linearly independent.
This follows from considering a linear combination which yields $0$, i.e. $\sum_{i=1}^{r+1} \lambda_i u_i=0,$ and studying the cases $\lambda_{r+1}\neq 0$ and $\lambda_{r+1}=0$ separately.
Applying the above to our collection of polynomials with increasing degrees, the result follows.
