Hello,

Ha ! I think I found it \o/

But no, g(s) is not necessarily equal to 0. I'm not too sure (I hate those limit/convergence theorems), but proving that may be enough (because both terms of the difference converge (*))

Consider the integral

Then divide the integral on intervals which boundaries are integers :

But

So :

Now :

But this is a telescoping sum !

So we have :

And this obviously goes to 0 as N goes to infinity (because )

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(*)

and this converges