I have an approximation operator $\hat I(f) = \sum A_i f(x_i)$ for $\int _a^bf$.
Prove that $\hat I$ has zero error for the interpolation polynomial of $f$ of degree $n$, iff it has zero error for any polynomial of degree at most $n$.
Now, it is trivial that if it has zero error for polynomials of degree at most $n$ then it has zero error for the interpolation polynomial, however I don't have a clue about the other direction, becuase I don't see why if it is exact for one polynomial of degree $n$ then it is exact for all of them. I can't just build from the elemantry functions $1, x, x^2, ...$ and use the linear combination because I am given accuracy for the whole polynomial. Isn't it possible that there are errors for parts of the polynomial (polynomials of smaller degrees), but the linear combination is exact?
Ok I understand now. Suppose the degree of some polynomial $P$ is $i<n$, then the interpolation polynomial of degree $n$ of $P$ is $P$, and so the approximation is exact.