Solution:
Call the expression $S$. Note that
$$\left(1+\frac{1}{2}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{6}\right) \ldots\left(1+\frac{1}{2018}\right)<\left(1+\frac{1}{1}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right) \ldots\left(1+\frac{1}{2017}\right)$$
Multiplying these two products together, we get
$$\begin{aligned} & \left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right) \ldots\left(1+\frac{1}{2018}\right) \\ & =\frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdots \frac{2019}{2018} \\ & =2019 \end{aligned}$$
This shows that
$$S^{2}<2019 \Longrightarrow S<\sqrt{2019}<50$$
as desired.