Let $d\in\mathbb N$ and $ P_d := \left\{p : K \to \mathbb{R} :\quad p(x) = \sum\limits_{i=1}^d p_i x^i\quad\text{ where }\quad \{p_i\}_{i=1}^d \subset \mathbb{R}\right\}, $ the set of all polynomials of degree at most $d$, where $K$ is a compact set. We have to show that $P_d$ isn't dense in the space of all continuous functions $C(K,\mathbb{R})$.
Our teacher gave us a hint: Consider a continuous function $f$ which has $d + 1$ zeros and show that all polynomials which approximate $f$ at its zeros with accuracy $\varepsilon > 0$ are uniformly bounded (Lagrange interpolation). Use this observation to show that there are continuous functions $f$ which cannot be approximated by $P_d$ with given accuracy $\varepsilon > 0$.
I can't figure out how to use this hint to solve the problem. Please help me out. Thanks.