Proving the collapse of a many body system (Fetter and Walecka problem 1.2)

I was trying to solve the problem 1.2 from Quantum theory of many-body systems by A. Fetter and J. D. Walecka. I succeeded in the first part, obtaining the suggested formulation for the expectation value of the Hamiltonian in the non-interacting ground state.

However, the problem asks to prove also that a system with a potential $V(|x_{1}-x_{2}|)<0 $ for every $|x_{1}-x_{2}|$ (attractive and central potential) would collapse and I can’t manage to do it. I wrote the mean energy per particle in terms of the Seitz parameter, but I’m stuck at this point:

begin{equation*}
frac{E^{0}+E^{1}}{N}= frac{3hbar^{2}}{10ma_{0}}(frac{9pi}{4})^{frac{2}{3}}frac{1}{r_{s}^{2}} +frac{3W(0)}{8pi a_{0}^{3}r_{s}^{3}}-frac{1}{2V}sum_{k m k’ m’}^{k_{F}} W(|k-k’|)
end{equation*}

where $W(k)$ indicates the Fourier transform of the potential and $r_{s}$ is the Seitz parameter.

I can’t figure out how to prove that the exchange term doesn’t create a repulsion at short range and, therefore, a minimum, causing a collapse of the system.

I add, as suggested, the text of the problem, to enhance the comprehension of the question:

Given a homogeneous system of spin-$frac{1}{2}$ particles interacting through a potential V:

(a) show that the expectiation value of the hamiltonian in the noninteracting ground state is

begin{multline*}
E^{0}+E^{1}= 2sum_{k}^{k_{F}} frac{hbar ^2 k^2}{2m} + +frac{1}{2}sum_{km,k’m’}^{k_{F}}[<km,k’m’|V|km,k’m’>-<km,k’m’|V|k’m’,km>]
end{multline*}

where m is the z-component if the spin.

(b) Assume $V$ is central and spin indipendent. If $V(|x_{1}-x_{2}|)<0$ for all $|x_{1}-x_{2}|$ and $int d^{3}x |V(x)|<infty$, prove that the system will collapse. (Hint: start from $frac{E^{0}+E{1}}{N}$ as function of denstity)