= Solution
\b[True.] By the <Weierstrass approximation theorem>, choose polynomials $q_n$ with $q_n\to f^{(m)}$ uniformly. Define $P_n$ by integrating $q_n$ exactly $m$ times and choose the $m$ integration constants so that
$$
P_n^{(r)}(0)=f^{(r)}(0),
\qquad 0\leq r<m.
$$
Then $P_n^{(m)}=q_n$, and repeated use of the <fundamental theorem of calculus> gives
$$
\|P_n^{(r)}-f^{(r)}\|_\infty
\leq\frac{\|q_n-f^{(m)}\|_\infty}{(m-r)!}
\qquad(0\leq r<m).
$$
Every derivative therefore converges uniformly as required.
Back to article page
