def bottom(root_ideal, start_index):
    """ Given a row index ``start_index``, look at it's :meth:`down_path` and return the final index.

        EXAMPLES:

        The picture below represents the root ideal used in the examples, and the path drawn on the picture depicts the down path for index 0 specifically, demonstrating that ``bottom(0)`` should be 4:

        .. image:: _static/bottom.JPG
            :width: 180px
            :align: center
            :alt: The root ideal [(0,2), (0,3), (0,4), (1,3), (1,4), (2,4)]

        ::

            sage: ri = RootIdeals().init_from_partition([3, 2, 1], 5)
            sage: ri.bottom(0)
            4
            sage: ri.bottom(1)
            3
            sage: ri.bottom(2)
            4
            sage: ri.bottom(3)
            3
            sage: ri.bottom(4)
            4

        ..  SEEALSO::

            :meth:`down`, :meth:`up`, :meth:`down_path`, :meth:`up_path`, :meth:`top`
        """
    return root_ideal.down_path(start_index)[-1]
