12.3.1.10 Algebraic

Condition: Prove that for arbitrary positive real numbers \( a_{1}, a_{2}, \ldots, a_{n} \) the following inequality holds: \[ \frac{n}{\frac{1}{1+a_{1}}+\frac{1}{1+a_{2}}+\cdots+\frac{1}{1+a_{n }}}-\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}} \geq 1\]