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cheer someone or something on
to encourage someone or a group to continue to do well, as by cheering. We cheered them on, and they won. We cheered on the team. Sam cheered Jane on.
Encourage, as in The crowd was cheering on all the marathon runners. Originating in the 1400s simply as cheer, this usage was augmented by on in the early 1800s.
To encourage someone with or as if with cheers: The spectators cheered the runners on as they passed by. I always cheer on the team that is losing.