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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 tothe regular continued fraction expansion of real numbers. In this paper weprove that the continued fraction converges to $F(x)$ for any positive realnumber $x>0$ by following the proof of that the regular continued fraction of apositive and irrational real number $\alpha$ converges to $\alpha$. Essentiallywe 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 anasymptotic relation between $Q_{2k}(x)$ and $Q_{2k-1}(x)$ by using propertiesof the classical Laguerre polynomial. In Section 7, we consider Euler-Gompertzconstant $\delta$. As far as we know, irrationality of $\delta$ is still anopen problem. We construct a sequence of rationals$\displaystyle\frac{A_{i}}{B_{i}}\ (i=1,2,3,\cdots)$ such that $\deltaB_{i}-A_{i}$ approaches 0 as $i$ approaches infinity and give a sufficientcondition 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 ofirrationality of Euler-Gompertz constant $\delta$.

Rational function approximations of the special function $e^{x}E_{1}(x)$ and applications to irrationality of Euler-Gompertz constant $\delta$