IMC 1995 · Problem 4

a) Prove that for every $\varepsilon > 0$ there is a positive integer $n$ and real numbers $\lambda_1, \dots, \lambda_n$ such that

$$\max_{x \in [-1,1]} \left| x - \sum_{k=1}^{n} \lambda_k x^{2k+1} \right| < \varepsilon.$$

b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\varepsilon > 0$ there is a positive integer $n$ and real numbers $\mu_1, \dots, \mu_n$ such that

$$\max_{x \in [-1,1]} \left| f(x) - \sum_{k=1}^{n} \mu_k x^{2k+1} \right| < \varepsilon.$$

Recall that $f$ is odd means that $f(x) = -f(-x)$ for all $x \in [-1,1]$.