Solution:

Label the numbers $a, b, c, d, e$ in increasing order. Place them around the pentagon in the order $e, a, d, c, b$. Then it is clear that the products of the numbers on the sides follow the inequalities
$$a d \leq a e \leq b e \quad \text{and} \quad b c \leq c d$$
Thus it suffices to prove that $b e \leq 1/9$ and $c d \leq 1/9$. Using the AM-GM inequality,
$$1 = a + b + c + d + e \geq 0 + 0 + c + d + d = c + 2d \geq \frac{3c}{2} + \frac{3d}{2} \geq 2 \sqrt{\frac{3c}{2} \cdot \frac{3d}{2}} = 3 \sqrt{c d}$$
so $c d \leq 1/9$. Also,
$$1 = a + b + c + d + e \geq 0 + b + b + b + e = 3b + e \geq 2 \sqrt{3b \cdot e}$$
so $b e \leq 1/12 < 1/9$.