Proof for Neumann boundary PDE

Question

Here is the problem :

Here is my attempt , Please have a look and point out any error:

Answer 1

It's correct, though at the moment it's more a list of equations rather than a proof - I'd link the lines after the first integral with "hence as $\partial u/\partial \nu=\nabla u \cdot \nu$", "so by the divergence theorem" and "as $-\Delta u = f$" respectively.