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Let a_0, a_1, \ldots, a_n, a_{n+1} be a sequence of real numbers satisfying the following conditions:

a_0 = a_{n+1 }= 0, |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).
Prove that |a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).

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