  We gave a method to construct a continued fraction of the function
$F(x):=e^{x}E_{1}(x)$. More precisely we define $F_{1}(x)$ as the reciprocal of
$F(x)$ and we inductively define $F_{m}(x)$ as the reciprocal of ``$F_{m-1}(x)$
minus the main term of $F_{m-1}(x)$ at infinity''. This method is analogous to
the regular continued fraction expansion of real numbers. In this paper we
prove that the continued fraction converges to $F(x)$ for any positive real
number $x>0$ by following the proof of that the regular continued fraction of a
positive and irrational real number $\alpha$ converges to $\alpha$. Essentially
we prove inequalities for $Q_{m}(x)$ (in Theorem 4.1) and inequalities
$F_{m}(x)>0$ (in Section 5). In particular, we prove stronger inequalities
$\displaystyle\frac{P_{2k}(x)}{Q_{2k}(x)}<F(x)<\displaystyle\frac{P_{2k-1}(x)}{Q_{2k-1}(x)}$
(than $F_{m}(x)>0$) and give two proofs of these. In Section 6, we show an
asymptotic relation between $Q_{2k}(x)$ and $Q_{2k-1}(x)$ by using properties
of the classical Laguerre polynomial. In Section 7, we consider Euler-Gompertz
constant $\delta$. As far as we know, irrationality of $\delta$ is still an
open problem. We construct a sequence of rationals
$\displaystyle\frac{A_{i}}{B_{i}}\ (i=1,2,3,\cdots)$ such that $\delta
B_{i}-A_{i}$ approaches 0 as $i$ approaches infinity and give a sufficient
condition of that $\delta B_{i}-A_{i}\neq 0$ for any positive integer $i$.
Therefore, if it is proved that this condition holds, it completes a proof of
irrationality of Euler-Gompertz constant $\delta$.
