Math Help - analytic function

1. analytic function

let $g:[0,1] \rightarrow R$ be a continuous function. Let $\epsilon >0$. Prove that there is a real analytic function $h: [0,1] \rightarrow R$ such that $|g(x)-h(x)| < \epsilon$ for all $x \in [0,1]$.

2. Originally Posted by Archi
let $g:[0,1] \rightarrow R$ be a continuous function. Let $\epsilon >0$. Prove that there is a real analytic function $h: [0,1] \rightarrow R$ such that $|g(x)-h(x)| < \epsilon$ for all $x \in [0,1]$.
You can even find a polynomial that will do this (that is the Weierstrass approximation theorem). A concrete way to construct a polynomial that approximates a given continuous function to any desired degree of accuracy is to use Bernstein polynomials.

3. can i just say it is proven by weierstrass approximation theorem? or do i have to actually show the proof ? i am not sure how to even start. whould u guide me?