Limit of exponential integral $ \lim{t \rightarrow \infty}{(te^t\int_{t}^{\infty}\frac{e^{-s}}{s}ds)} $

Limit of exponential integral $ \lim{t \rightarrow
\infty}{(te^t\int_{t}^{\infty}\frac{e^{-s}}{s}ds)} $

Evaluate the following limit: \begin{equation} \lim\limits_{t \rightarrow
\infty}{(te^t\int\limits_{t}^{\infty}\frac{e^{-s}}{s}ds) := I}
\end{equation}



That's the Exponential integral, which is not an elementary function.
Considering $ \frac{1}{s} < \frac{1}{t} $ we get $ I \le 1 $, and my
intuition says it's actually the answer ($ \lim{I} = 1 $), but I can't
prove that $ I \ge 1 $. Am I right?