There exists a segment such that . Choose to be a -valued function that equals 1 on and 0 on where (for instance). Then, since , using the assumption there exists such that, for , .
Now write . We have . Same with respect to , hence .
And for the first term, Stone-Weierstrass provides a polynomial such that for , hence , and is in . Hence , and we also have for using the assumption. Finally, like with .
Glue together the above pieces, and you get what you want for :