Recently, I came across Huygens inequality from here which stated that:
If $a_{1}, a_{2}\ldots a_{n}$ are $real\ numbers$, then $$ \left(1 + a_{1}\right)\left(1 + a_{2}\right)\ldots\left(1 + a_{n}\right) \geq \left[\vphantom{\large A^{A}}1 + \left(a_{1}\,a_{2}\,a_{3}\ldots a_{n}\right)^{1/n}\,\right]^{n} $$ I was trying to prove this inequality using mathematical induction. But I can't find a relation between $n$ and $n + 1$. I also couldn't find any proof for the same on internet. How can we go about proving this inequality using simple or strong induction or am I going in wrong direction altogether $?$.
