Show that for reals $>0$ $a_1,a_2,...,a_n$ : $a_1a_2...a_n < 1 \iff$ $\exists k \in \mathbb{N}$ $\geq 2$ such that $\sum_{j=1}^n \sqrt[k]{a_i} < n$

Question

Show that for any positive reals $a_1,a_2,...,a_n$ we have $a_1a_2...a_n < 1 \iff \exists k \in \mathbb{N}, k\geq 2$ for which $\sqrt[k]{a_1} + \sqrt[k]{a_2} + ... + \sqrt[k]{a_n} < n$.

The $\Leftarrow$ part is very easy (just apply AM-GM for that $k$). I wasn't able to solve the other part.

It's enough to show that if $\sqrt[k]{a_1} + \sqrt[k]{a_2} + ... + \sqrt[k]{a_n} \geq n$ for any $k$, then $a_1a_2...a_n \geq 1$. Any ideas?