For arbitrary real numbers \( x_1,x_2,\ldots,x_n \), prove that \[ \left(\max_{1\leq i \leq n}x_i \right)^2 + 4\sum_{i=1}^{n-1}\left(\max_{1\leq j \leq i}x_j\right)\left(x_{i+1}-x_i\right) \leq 4x_n^2. \]