Inverse laplace transform pf infinite product !

[Maszenka]

I have to do inverse laplace transform of infinite product that is shown below. Can somebody help me with that?

 

[romsek]

\(\displaystyle g(t) =\dfrac{ e^{-nt}}{(n-1)!} (e^t-1)^{n-1}\)

I suspect it can be proven using induction. Have a go.