Let $C[0,1]$ denote all the real-valued continuous function on $[0,1].$ Consider the normed linear space
$$X = \left \{f \in C[0,1] : f \left ( \frac 1 2 \right ) = 0 \right \},$$
with the sup-norm$,$ $\|f\| = \sup\ \{|f(t)| : t \in [0,1] \}.$ Show that the set
$$P = \left \{f \in X : f\ \text {is a polynomial} \right \}$$
is dense in $X.$
I take a basic open set $B(f,\epsilon) \cap X$ in $X,$ where $f \in X$ and $\epsilon > 0.$ Now by Stone-Weierstrass theorem there exists a sequence of polynomials in $C[0,1]$ converging uniformly to $f$ i.e. $\|f_n - f \| < \epsilon,$ for all $n \geq k.$ So if I can find an $n \geq k$ such that $f_n \left (\frac 1 2 \right ) = 0$ then the proof is complete.
How can I find such an $n$? Please help me in this regard.
Thank you very much.