Question 4
Let $a_0,a_1,a_2,\ldots,a_n,a_{n+1}$ be a finite sequence of real numbers satisfying
$$a_0=a_{n+1}=0,$$
$$a_{k-1}-2a_k+a_{k+1}\le1\qquad(k=1,2,\ldots,n).$$
Prove that for $k=0,1,\ldots,n+1$,
$$a_k\le\frac{k(n+1-k)}2.$$
$$a_0=a_{n+1}=0,$$
$$a_{k-1}-2a_k+a_{k+1}\le1\qquad(k=1,2,\ldots,n).$$
Prove that for $k=0,1,\ldots,n+1$,
$$a_k\le\frac{k(n+1-k)}2.$$