Equivalence of Definitions of Euler-Gompertz Constant

Integral Form implies Continued Fraction Form
Let $G$ be the integral form of the Euler-Gompertz constant.

Then by definition:
 * $G = \ds \int_0^\infty \dfrac {e^{-u} } {1 + u} \rd u$

Thus $G$ equals the continued fraction form of the Euler-Gompertz constant.

Continued Fraction Form implies Integral Form
Let $G$ be the continued fraction form of the Euler-Gompertz constant.

Then by definition:
 * $G = \cfrac 1 {2 - \cfrac {1^2} {4 - \cfrac {2^2} {6 - \cfrac {3^2} {8 - \cfrac {4^2} {10 - \dotsb} } } } }$

Thus $G$ equals the integral form of the Euler-Gompertz constant.