Finite element methods proof(help asap please)

greek9000

Hi there I'm having a trouble with the following proof.I have the exam soon and he has asked this question every year so I have no doubt that he will ask it again.I have no idea how to do it?If you can help me I would be greatly appreciative.

Question:
Consider the 2-point boundary value problem
-u'' = f(x) on (0,1)
subject to the boundary conditions u(0) and u'(1)=0. The associated weak formulation reads:Find u element of V such that a(u,v) = l(v) for all v is an element of V,where a(u,v) = ∫(0 to 1) u'v'dx and l(v) = ∫(0 to 1) f(x)v(x)dx

Then:
(1)Define the suitable function space V(1)
(2) prove that if f is an element of C[0,1] and u is an element of C^2[0,1] satisfies the variational formulation,then u solves the differential equation.(7)
(3)Define the Ritz-Galerkin approximation to the variational statement.(2)
(4)Prove that if f is an element of L(subscript 2 at the bottom)(0,1),then the solution u(subscript h at the bottom) is an element of Vh(subscript h at the bottom of V) such that a(uh,vh) = (f,vh) for all vh is an element of Vh is unique? (8)

I need help with question 4 asap please guys.I'm desperate.Stressing out big time. Thanks very much in advance.