Increasing every starting edge by a constant, then the shortest path tree remains the same?
I think you can prove it by counter-example.

Assume your SPT (shortest path tree) from the original graph is no longer a valid SPT in the new graph.  That means there's a node $f$ s.t. the path $(s, n_1, n_2, ..., f)$ in your SPT is no longer a shortest path, meaning some other path $(s, v_1, v_2, ..., f)$ (whose length wasn't shorter before) now has a shorter length.

Now show that's impossible by calculating how much the lengths of those two paths have increased.