19.5.2 Compactness

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