19.5.2 Compactness

Problem: Prove that in the space $C[0,2]$ the following set is precompact $$A=\{f\in C[0,2]||f^{\prime \prime }(x)|\le 3,\mathrm{\forall }x\in [0,2];|f^{\prime }(0)\mid \le 1,f(2)=3\}\text{. }$$