On-line antichain partition of a poset of bounded height

Statement

Consider the on-line game in which the points of a partially ordered set $P$ are shown to Assigner one at a time. When a point appears, Assigner sees how it compares with the points shown before it. She must at once put it into a class, and she can never move it later. Each class must end up as an antichain.

Fix an integer $h \ge 1$. Then Assigner has a strategy for this game that works for every poset $P$ whose height is at most $h$. This strategy splits $P$ into at most $\binom{h+1}{2}$ antichains.