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