Can someone show me how the space of polynomials which take [a,b] to R are a "thin" subset of C0[a,b], i.e. the space of continuous functions from [a,b] to R.

Note: A "thin" subset is sometimes also called a "meager" subset and basically means that P can be written as the countable union of closed and nowhere dense sets:

$\displaystyle P = \bigcup_{k=1}^{\infty}H_k $ where $\displaystyle H_k $ is closed and nowhere dense.