Prove limit is the Euler-Mascheroni constant $\gamma$

Question

How do I prove this result?

$$\lim_{\varepsilon\to0^+}\left[\operatorname{li}(1-\varepsilon)-(1-\varepsilon)\ln\ln\frac{1}{1-\varepsilon}\right]=\gamma$$with$$\operatorname{li}(x):=\int_0^x\frac{dt}{\ln t}$$

Wolfram Alpha obtains this limit.