Let $f,g\in C[0,1]$ such that $f(x)<g(x)$ for all $x\in [0,1]$. Show that there exists a polynomial $p\in C[0,1]$ such that $f(x)<p(x)<g(x)$ for all $x\in [0,1]$.
I know that I have to somewhere use Weierstrass Approximation Theorem in order to prove the above statement but I really am not able to get the idea. Please someone give me any hint or something.
Yup, let $$h(x)=\frac{f(x)+g(x)}{2},\, \varepsilon=\frac{\min|f(x)-g(x)|}{2},$$ then apply Weierstrass approximation theorem for $h(x),\,\varepsilon$.