Non Compact Amenable group does not have Property (FH)

Question

I'm learning from some notes about Analytic Group Theory and I'm stuck on an exercise there.

Prove that a non compact Amenable group does not have Property (FH).

At this point of the notes we learned about Amenability and know the following equivalence:

1. $G$ is amenable (has a finite additive, finite left (or right or both) invariant measure defined on all subsets.

2. $G$ has a left (or right or both) invariant mean.

3. $G$ admits almost invariant functions in $L^p(G)$.

4. Any Hausdorff Compact space such that $G$ acts on it continuously admits a $G$-invariant Borel measure.

5. Any compact convex set such that $G$ acts on it with affine transformations has a fixed point.

The only thing we know about (FH) property is the definition: A group has property (FH) if any isometric action of it on a Hilbert space admits a fixed point.

Later on in the notes it is proven that property (T) and property (FH) are (almost) equivalent. I don't want to use that (I think that's cheating because the exercise was given before).

Thank you