$\begin{align}  & \left( a+b+c+d \right)\left( \tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}+\tfrac{1}{d} \right)=\left( 1+\tfrac{a}{b}+\tfrac{a}{c}+\tfrac{a}{d}+\tfrac{b}{a}+1+\tfrac{b}{c}+\tfrac{b}{d}+\tfrac{c}{a}+\tfrac{b}{c}+1+\tfrac{d}{c}+\tfrac{d}{a}+\tfrac{d}{b}+\tfrac{d}{c}+1 \right) \\ & \left( a+b+c+d \right)\left( \tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}+\tfrac{1}{d} \right)=\left[ 4+\left( \tfrac{a}{b}+\tfrac{b}{a} \right)+\left( \tfrac{a}{c}+\tfrac{c}{a} \right)+\left( \tfrac{a}{d}+\tfrac{d}{a} \right)+\left( \tfrac{b}{c}+\tfrac{c}{b} \right)+\left( \tfrac{b}{d}+\tfrac{d}{c} \right)+\left( \tfrac{c}{d}+\tfrac{d}{c} \right) \right] \\ & \text{note that }\tfrac{x}{y}+\tfrac{y}{x}\ge 2 \\ & \left[ 4+\left( \tfrac{a}{b}+\tfrac{b}{a} \right)+\left( \tfrac{a}{c}+\tfrac{c}{a} \right)+\left( \tfrac{a}{d}+\tfrac{d}{a} \right)+\left( \tfrac{b}{c}+\tfrac{c}{b} \right)+\left( \tfrac{b}{d}+\tfrac{d}{c} \right)+\left( \tfrac{c}{d}+\tfrac{d}{c} \right) \right]\ge 4+2+2+2+2+2+2 \\ & \left( a+b+c+d \right)\left( \tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}+\tfrac{1}{d} \right)\ge 16 \\\end{align}$