Let $\{a_k\}_{k\ge 0}, \{b_k\}_{k\ge 0}, \{\xi_k\}_{k\ge 0}$ are non-negative sequences, for all $k\ge 0$, $a_{k+1}^2 \le (a_k + b_k)^2 - \xi_k^2$.
(1) Prove that $\sum_{i=1}^k \xi_i^2 \le \left( a_1 + \sum_{i=0}^k b_i \right)^2$.
(2) If $\{b_k\}_{k\ge 0}$ satisfy that $\sum_{k=0}^{\infty} b_k^2 < +\infty$, then $\lim\limits_{k\to \infty} \frac{1}{k}\sum_{i=1}^k \xi_i^2 = 0$.
Any hints are appreciated, thanks for your help.