Give a detailed proof of the assertion that the Weierstrass theorem for a general $[a,b]$ follows from the result on $[0,1]$ (by using lemma 11.1)
It's been two days since I am trying to figure out the way of proving this problem in (N.L. Carothers), any hints please?
Hint: Start with $g \in C[a,b]$, from Lemma 11.1, there is $f \in C[0,1]$ such that $g(x) = f(t(x))$ where $t(x) = \frac{x-a}{b-a} \in [0,1]$.
Do the logical steps till you get:
$$\lVert g(x) - p_n (x) \rVert = \max_{a\le x \le b} |f(t(x)) - P_n(t(x))|=\max_{0\le t \le 1} |f(t) - P_n(t)| < \epsilon.$$
That is because we have already know that the space of polynomials is dense in $C[0,1]$.