Existence of degree $n$ best approximation polynomial implies Existence of degree $n+1 $ best approximation polynomial

There exist a degree $n+1$ polynomial of best approximation if there exist a degree $n$ polynomial of best approximation

This is the context of the question.

If there exists a polynomial of best approximation of degree n, there also exists a polynomial of best approximation of degree n+1.

This is another question been asked with same purpose as this one that hasn't been answered. That is, for a method of solution that is relevant to the text's level, the problem is from an elementary analysis text Zorich Mathematical Analysis I Chapter 4.

So my problem is that just like the second link, I need a rather elementary solution to this problem, which I assume can be achieved, but have not been achieved myself.

Any help is greatly appreciated.