integration - Limit of the Exponential Integral

I want to show the following (I am only really interested in real variables/parameters, so $x,b,c\in\mathbb{R}$):

$$\lim_{x\rightarrow\infty}E_1\left[b\left(x+c\right)\right]=0,\quad\text{for }b>0$$

Where I want to use the definition of the exponential integral:

$$E_1\left(x\right)=\int^{\infty}_{x}\frac{e^{-t}}{t}dt,\quad(x>0)$$

I am not sure how to show this properly. If I try to just plug in the definition, I believe I can see the expected result if I exchange the limit and the integration - but being a physicist I don't really know why I would be allowed to. I read about the dominated convergence theorem, but don't understand if/how this can be applied. Can someone give me a mathematically correct (and in the best case easy to understand) derivation?

Thanks for any hints in advance.

Answer

Think of $E_1(n)$ for $n\ge 1$ as
$$\int_1^\infty 1_{[n,\infty]}(t){e^{-t}\over t}\,dt$$
where $1_{[n,\infty]}$ denotes a function that is $1$ on $[n,\infty]$ and $0$ otherwise. The integrand for all $n\geq1$ is dominated by $e^{-t}/t$ (which is positive and has a finite integral). The integrands converge pointwise to $0$ as $n\to \infty$. The integrals thus converge to zero, and so $\lim_{n\to\infty}E_1(n)=0$. Since $E_1(x)$ is decreasing as a function of $x$ on account of the integrand being positive and since $b>0$, it follows that $\lim_{x\to\infty}E_1(b(x+c))=0$.