Proof: Assuming $a \leq b \leq c$, according to Lemma 5, we infer that $a^{2}+b^{2}+c^{2}$ reaches its maximum value if and only if $a=0$ or $b=c=2$. In the first case, $a=0$ is impossible, because $4 \geq b+c=5$, leading to a contradiction. In the second case, we have $a=1$, thus $\max \left\{a^{2}+b^{2}+c^{2}\right\}=1^{2}+2^{2}+2^{2}=9$