Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $f$ be a real-valued function of class $\mathcal{C}^n$ on $I$ and $P = \Pi(f)$ its Lagrange interpolation polynomial. Deduce that
$$\sup _ { x \in [ a , b ] } | f ( x ) - P ( x ) | \leqslant \frac { M _ { n } ( b - a ) ^ { n } } { n ! }$$
where $M _ { n } = \sup _ { x \in [ a , b ] } \left| f ^ { ( n ) } ( x ) \right|$.