I am trying to complete the following proof:
for $ 0 < \epsilon < 1 $, show that:
$ ln(\frac{1}{\epsilon}) \leq e \int_{\epsilon}^{\infty} \frac{exp(-x)}{x} dx $
What I am currently trying to do is get the left hand side in terms of an integral whose integrand is bounded by that on the right, but this isn't working too well.
Note: This is not explicitly stated as an incomplete gamma function, however I noted that it was appeared to be one in case there was some accompanying theory of gamma functions that could help with such a proof.
Any help, even a hint as to where to start, would be much appreciated.
Let $f(t)=e\int_t^{\infty} \frac {e^{-x}} x dx+\ln t$. Then $f'(t)=-e \frac {e^{-t}} t+\frac 1 t$. Since $e^{1-t} \geq 1$ for $0<t<1$ we see that $f$ is decreasing on $(0,1]$ Since $f(1) \geq 0$ we see that $f(t) \geq 0$ for $0<t<1$ and this is equivalent to the given inequality.