The claim is 
>!**false**.

>!Choose a diameter, and draw a point at each end.

>!If neither point is on the dividing equator, they must lie in opposite hemispheres.

>!If you have two such pairs, each hemisphere must have at least 2 points.

>!Since there are only 5 points total, each hemisphere can have at most 3 points.

Edit:
>! If we allow that points can lie on the dividing equator, the above will not suffice as a counterexample.  See the comments below.