Prove that $|x|$ can be uniformly approximated in [-1,1] by polynomials?

Question

I know how to aproximate it, is it enough to actually aproximate it to prove it can be aproximated?

I just noticed there is a suggestion here at the end. Sorry for lateness

Do Taylor arround the funcion $f(t):=t\sqrt{1-t}$ and $t\in [-1,1]$ Show that $f\in C^\infty$ in $[-1,1)$ and that its taylor serioes is uniformly convergent in compacts of $[-1,1)$ Its derivate tends to $\infty$ if $t\rightarrow +1$ but f is continuous, then it is uniformly continuous in $ [-1,+1]$.

Not sure which theorem or result should be used here.

Answer 1

With Stone-Weierstrass theorem, since $|x|$ is continuous on a closed inteval, it can be approximated uniformly by polynomials. If you want an example of such an approximation of your function, you can use Berstein's polynoms :$$B_n(f)=\sum_{k=0}^n f\left(\frac{k}{n}\right)\binom{n}{k}x^k(1-x)^{n-k}$$

Answer 2

You'd need to prove that the approximation you find is an approximation.