Induction Proof from Thomas Judson book on abstract algebra

Question

I'm trying to prove $$^n\sqrt{a_1\times a_2\times...\times a_n}\leq \frac{1}{n}\sum_{k=1}^na_k, \quad a_i\in \mathbb{Z}^+$$ by Induction. The case is true for $n=1$ so I assumed true for $n=k$. I then tried \begin{align*} ^{k+1}\sqrt{a_1\times a_2\times ...\times a_{k+1}} &\leq \enspace ^{k}\sqrt{a_1\times a_2\times...\times a_k}\times ^{k}\sqrt{a_{k+1}} \\ &\leq \frac{1}{k}\sum_{i=1}^{k}a_i \times ^{k}\sqrt{a_{k+1}} \end{align*} and have tried playing around with the algebra to try and get the desired result but cannot get it to work. Any hints would be very much appreciated.