It is intuitive that all polynomials up to and including order $p$ can be fully recovered i.e. without error, but how can one rigorously prove this?
In the book by Lambert, there is a similar question-
Let $L$ be the linear difference operator associated with a linear multistep method. Show that the method has order $p$ if and only if
$$ L[x^r;h]=0, r=0,1,...p $$