lose heart
Become discouraged, as in The rescuers worked hard for the first few hours, but then they lost heart. This term uses heart in the sense of “courage” or “spirit.” [Mid-1800s] Also see lose one's heart to.
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The American Heritage® Idioms Dictionary
Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.