(Solved): Use mathematical induction to prove the following statements. (6 points (e) For any \( n \in \math ...

Use mathematical induction to prove the following statements. (6 points (e) For any \( n \in \mathbb{N} \backslash\{0\} \), \[ \sum_{r=1}^{n} \frac{1}{r^{2}} \leq 2-\frac{1}{n} . \] (f) Let \( a_{1}=1 \) and \( a_{2}=1 \). Define \( a_{n}=a_{n-1}+a_{n-2} \) when \( n \geq 3 \). Then for all \( n \in \mathbb{N} \backslash\{0\} \), \[ \sum_{r=1}^{n} a_{r}=a_{n+2}-1 . \] (g) Let \( f(x)=\frac{1}{1+x^{2}} \). Then for any \( n \in \mathbb{N} \), \[ \left(1+x^{2}\right) f^{(n+2)}(x)+2(n+2) x f^{(n+1)}(x)+(n+2)(n+1) f^{(n)}(x)=0 . \]