I've encountered the following statement:

Let $a, b_1, \ldots, b_m\in P([0, 1])$ be polynomials over $[0, 1]$. If $\mathrm{span}\{a^k b_j, j=1, \ldots, m, k = 0, 1, \ldots\}$ is uniformly dense in $C([0, 1])$, then $\mathrm{span}\{(a^k b_j)', j = 1, \ldots, m, k = 0, 1, \ldots\}$ is dense in $L^2([0, 1])$ with respect to the $L^2$-norm.

I guess the statement is false. But I can't figure out a counter-example. Can anyone give me some idea of proving or disproving the statement?